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).