Calculate geometrically, the area bounded by the following functions and the x-axis over the given domain:
2 \leq x \leq 5
0 \leq x \leq 7
0 \leq x \leq 4
0 \leq x \leq 9
Calculate geometrically, the exact value of the following definite integrals:
\int_{0}^{6} \left(6 - x\right) dx
\int_{0}^{12} f \left( x \right) dx
\int_{0}^{16} f \left( x \right) dx
\int_{1}^{8} \left(10 - x\right) dx
\int_{3}^{5} \left( 2 x + 1\right) dx
\int_{0}^{16} f \left( x \right) dx
\int_{ - 6 }^{6} \sqrt{36 - x^{2}} dx
\int_{0}^{6} \sqrt{36 - x^{2}} dx
\int_{0}^{16} f \left( x \right) dx
\int_{0}^{12} \sqrt{36 - \left(x - 6\right)^{2}} dx
For each of the following functions:
Sketch a graph of f \left( x \right).
Hence, calculate geometrically the area bounded by the curve and the x-axis over the given domain:
f \left( x \right) = 6 - 6 x for 0 \leq x \leq 1
f \left( x \right) = 3 x + 4 for 0 \leq x \leq 3
f(x)=\begin{cases} x & \text{for } 0\leq x \lt 4\\ 8-x & \text{for } 4\leq x\leq 8 \end{cases}
f(x)=\begin{cases} 2x & \text{for } 0\leq x \leq 3\\ 6 & \text{for } 3\lt x\lt 6\\ 18-2x & \text{for } 6\leq x \leq 9 \end{cases}
Consider the function f \left( x \right) = x^{2} - 3 x + 10.
For each of the following intervals, state whether the right endpoint approximation method will underestimate or overestimate the area under the curve. Explain your answer.
\left[ 3, 8 \right]
\left[ -6, -2 \right]
Consider the function f \left( x \right) = x^{2} + 5.
For each of the following intervals, state whether the left endpoint approximation method will underestimate or overestimate the area under the curve. Explain your answer.
\left[ 2, 6 \right]
\left[ -5, -1 \right]
Consider the function f \left( x \right) = 10 e^{x}.
State the approximation method that will give an underestimate of the true area on any given interval. Explain your answer.
Consider the function f \left( x \right) = -3 x^{2} - 24 x + 1.
State whether the right endpoint approximation to the area will overestimate or underestimate on the following intervals.
\left[ -10, -5 \right]
\left[ -4, 2 \right]
\left[ 0, 2 \right]
\left[ -6, -4 \right]
State whether the left endpoint approximation to the area will overestimate or underestimate on the following intervals.
\left[ -10, -5 \right]
\left[ -4, 2 \right]
\left[ 0, 2 \right]
\left[ -6, -4 \right]
The function f \left( x \right) = 5 x is defined on the interval \left[0, 6\right].
Graph f \left( x \right).
Find the area under f \left( x \right) by partitioning \left[0, 6\right] into 3 sub-intervals of equal length using:
Left endpoint approximation
Right endpoint approximation
Find the area under f \left( x \right) by partitioning \left[0, 6\right] into 6 sub-intervals of equal length using:
Left endpoint approximation
Right endpoint approximation
Find the actual area under the curve on the interval \left[0, 6\right].
The function f \left( x \right) = - 4 x + 12 is defined on the interval \left[0, 3\right].
Graph f \left( x \right).
Find the area under f \left( x \right) by partitioning \left[0, 3\right] into 3 sub-intervals of equal length using:
Left endpoint approximation
Right endpoint approximation
Find the area under f \left( x \right) by partitioning \left[0, 3\right] into 6 sub-intervals of equal length using:
Left endpoint approximation
Right endpoint approximation
Find the actual area under the curve on the interval \left[0, 3\right].
The interval \left[0, 8\right] is partitioned into 4 sub-intervals \left[0, 2\right], \left[2, 4\right],\left[4, 6\right], and \left[6, 8\right].
Find the area under the curve on the interval \left[0, 8\right] using:
Left endpoint approximation
Right endpoint approximation
Use left endpoint approximation to find \int_{1}^{9} 2 x^{2} dx by using 4 rectangles of equal width.
Use midpoint approximation to find the following by using 4 rectangles of equal width:
Give your answer to one decimal place if necessary.
\int_{0}^{8} 8 x \ dx
\int_{1}^{3} \left(4 - x\right) dx
\int_{ - 2 }^{2} \left(e^{x} + 1\right) dx
Use right endpoint approximation to find \int_{1}^{5} \dfrac{1}{x} dx by using 4 rectangles of equal width.
Give your answer as a simplified fraction.
Approximate \int_{3}^{15} \left( 4 x + 6\right) dx by using 4 rectangles of equal width and using the method:
Midpoint approximation
Left endpoint approximation
Right endpoint approximation
A circle with centre at the origin and radius of 16 units has equation x^2+y^2=256.
What fraction of the area of the whole circle does the integral \int_{0}^{16} \sqrt{256 - x^{2}} \ dx represent?
Use midpoint approximation to find the integral by using 4 rectangles of equal width. Round your answer to one decimal place.
Find the exact value of the integral by using the formula for the area of a circle.
Hence find the difference between the approximate and exact values of the integral. Round your answer to one decimal place.
Use technology to find the exact value of the following definite integrals:
Consider the function f \left( x \right) = 0.5 x^{2}.
Complete the table to estimate the area between the function and the x-axis for \\1 \leq x \leq 3 using the left endpoint approximation method. Round your answers to three decimal places.
| n | 5 | 10 | 100 | 1000 | 10\,000 |
|---|---|---|---|---|---|
| A_L \text{ units}^2 |
Use technology to evaluate \int_{1}^{3} 0.5x^2 \ dx and hence confirm that the exact area is the limit of A_L as n gets larger.
Consider the function f \left( x \right) = \sin x.
Complete the table to estimate the area between the function and the x-axis for \\0 \leq x \leq \pi using the left endpoint approximation method. Round your answers to three decimal places.
| n | 5 | 10 | 100 | 1000 | 10\,000 |
|---|---|---|---|---|---|
| A_L \text{ units}^2 |
Use technology to evaluate \int_{0}^{\pi} \sin x \ dx and hence confirm that the exact area is the limit of A_L as n gets larger.
Consider the function f \left( x \right) = \sqrt{4 - x^{2}}.
Complete the table to estimate the area between the function and the x-axis for \\0 \leq x \leq 2 using the right endpoint approximation method. Round your answers to five decimal places.
| n | 5 | 10 | 100 | 1000 |
|---|---|---|---|---|
| A_R \text{ units}^2 |
Use technology to evaluate \int_{0}^{2} \sqrt{4 - x^{2}} \ dx and hence confirm that the exact area is the limit of A_R as n gets larger.
Consider the function f \left( x \right) = - x^{2} + 3 x.
Sketch a graph of f(x).
State the x-values that define the region bounded by the curve and the x-axis. Write your answer as an inequality.
Complete the table to estimate the area of the region bounded by the function and the \\x-axis using the right endpoint approximation method. Round your answers to four decimal places.
| n | 5 | 10 | 100 | 1000 | 10\,000 |
|---|---|---|---|---|---|
| A_R \text{ units}^2 |
Use technology to confirm that the exact area is the limit of A_R as n gets larger.
Consider the function f \left( x \right) = e^{x}, where x > 0:
Use technology to complete the table by finding the exact area between the function and the x-axis for 0 \leq x \leq a.
| \text{Value of }a | 1 | 2 | 3 | \dfrac{1}{2} | \dfrac{1}{3} |
|---|---|---|---|---|---|
| \text{Integral} \int_{0}^{a} e^x dx | \int_{0}^{1} e^x dx | \int_{0}^{2} e^x dx | \int_{0}^{3} e^x dx | \int_{0}^{\frac{1}{2}} e^x dx | \int_{0}^{\frac{1}{3}} e^x dx |
| \text{Exact area (A)} |
Hence find a rule for the area A, between the graph of f \left( x \right) = e^{x} and the x-axis over the interval 0 \leq x \leq a, for a > 0.
Use the rule found in part (b) to determine the exact the area between the graph of f \left( x \right) = e^{x} and the x-axis over the interval 0 \leq x \leq \pi.
Find an expression for the indefinite integral g\left(x\right)=\int e^x \ dx.
Show that g\left(\pi\right)-g\left(0\right) is equal to the area found in part (c).
Consider the following functions of the form f \left( x \right) = x^{a}, where a > 0:
Use technology to complete the table by finding the exact area between the function and the x-axis for 0 \leq x \leq 1.
| \text{Value of }a | 2 | 3 | 1 | \dfrac{1}{2} | \dfrac{1}{3} |
|---|---|---|---|---|---|
| \text{Function }f(x) | x^2 | x^3 | x | x^{\frac{1}{2}} | x^{\frac{1}{3}} |
| \text{Area (A)} |
Hence find a rule for the area A, between the graph of f \left( x \right) = x^{a} and the x-axis over the interval 0 \leq x \leq 1, for a > 0.
Use the rule found in part (b) to determine the exact the area between the graph of f \left( x \right) = x^{\pi} and the x-axis over the interval 0 \leq x \leq 1.
Find an expression for the indefinite integral g\left(x\right)=\int x^a \ dx, for a > 0.
Show that g\left(1\right)-g\left(0\right) is equal to the rule found in part (b).
Use the trapezium rule to find an approximation for each integral:
Find an approximation to \int_{0}^{1}x^{3} \, dx using the trapezium rule with:
1 subinterval
2 subintervals
Given the table of values, find the approximate value of each definite integral:
\int_{0}^{8}f \left(t\right) \, dt
| t | 0 | 2 | 4 | 6 | 8 |
|---|---|---|---|---|---|
| f\left( t \right) | 3 | 7 | 11 | 9 | 3 |
\int_{0}^{40}f \left(x \right) \, dx
| x | 0 | 10 | 20 | 30 | 40 |
|---|---|---|---|---|---|
| f\left( x \right) | 350 | 410 | 435 | 450 | 460 |
\int_{-4}^{2}f\left(x\right) \,dx
| x | -4 | -3 | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|---|---|
| f\left( x \right) | 0 | 4 | 5 | 3 | 10 | 11 | 2 |
Use the trapezium rule with 2 subintervals to find an approximation for the following to three decimal places:
Find an approximation to the following integrals, rounding your answers to three decimal places:
\int_{2}^{3}e^{2x} \,dx using 3 subintervals
\int_{0}^{2} \left(4x^{2}-16 \right) \,dx using 4 subintervals
\int_{0}^{1}\sqrt{2x}\,dx using 5 subintervals
\int_{1}^{3}\dfrac{dx}{x^{3}} using 4 subintervals
\int_{2}^{5} \dfrac{2}{3x-4}\,dx\,using 6 subintervals
Use one application of the trapezium rule to approximate \int_{2}^{3} \dfrac{e^{x}}{x}\, dx to one decimal place.
Use three applications of the trapezium rule to approximate \int_{0}^{9} e^{ - x^{2} }\, dx to one decimal place.
Approximate \int_{0}^{8} 8 x \, dx by using four rectangles of equal width whose heights are the values of the function at the midpoint of each rectangle.
Approximate \int_{1}^{5} \dfrac{1}{x} \, dx by using four rectangles of equal width whose heights are the values of the function at the right endpoint of each rectangle.
In the following graph, the interval \left[0, 8\right] is partitioned into four subintervals \left[0, 2\right], \left[2, 4\right], \left[4, 6\right], and \left[6, 8\right]:
Approximate the area A using rectangles for each subinterval whose heights are equal to the function values of the left side of the subintervals.
Approximate the area A using rectangles for each subinterval whose heights are equal to the function values of the right side of the subintervals.
The function y = 3 \ln x has been graphed:
Use two applications of the trapezium rule to approximate the area bound by the curve, the x-axis and and x = 6. Round your answer to one decimal place.
Consider the function y = e^{x^{2}}.
Determine y''.
Using two applications of the trapezium rule, approximate the area bound by the curve and the x-axis between x = 0 and x = 2. Round your answer to one decimal place.
Is the approximation given by the trapezium rule an underestimate or an overestimate of the actual area? Explain your answer.
Consider the integral \int_{1}^{5} \left(\ln x + 4\right) \, dx.
Approximate the area bounded by the curve, the x-axis, x = 1 and x = 5 using two applications of the trapezium rule. Round your answer to two decimal places.
Use the trapezium rule to find the approximate area of the irregular figure below:
The following piece of land has straight boundaries on the east, west and south borders and is bounded by a creek to the north. The land has been divided into two sections so we can use the trapezium rule to approximate the area:
Find the approximate area of the piece of land by using two applications of the trapezium rule.
During a heavy storm, 35.2 \text{ mm} of rain fell. Find the volume of water that falls on this land. Round your answer to the nearest cubic metre.
A surveyor made the following diagram with measurements for a property she was mapping out. On the west side of the property is a river:
Find the approximate total area of the property by using three applications of the trapezium rule.
The average weekly rainfall is 34 \text{ mm}. Calculate the total volume of water that falls on the land in cubic metres. Round your answer to two decimal places.
A river has its depths marked out at equal intervals of 9 \text{ m}. The depths are 0, 12, 14, 17, 5, and 0 \text{ m} respectively. Find the approximate area of the cross section of the river.
The elevation values of a mountain are recorded at equal intervals of 250 \text{ m}. The heights are shown in the diagram:
Find the approximate area of the cross section of the mountain.
The diagram shows the cross-section of a river. The depths of the river are marked at 2-metre intervals:
Using three applications of the trapezium rule, approximate the area of the cross-section of the river to one decimal place.
Use four applications of the trapezium rule to approximate the area of the cross-section of the following river:
The following shape has measurements given in metres. Use the trapezium rule to find the area in hectares.
A garden is 49 \text{ m} long. At 7 \text{ m} intervals, the width of the garden was given by the following measurements:
0 \text{ m}, \, 2.9\text{ m}, \, 5.2\text{ m}, \, 6.6\text{ m}, \, 5.6\text{ m}, \, 4.3\text{ m}, \, 3\text{ m}, \, 2.5\text{ m}
Using the trapezium rule, approximate the area of the garden to two decimal places.