14.01 Approximation of areas under a curve
The following graph shows the function y= e^{x}:
Find an estimate for the area between the function and the x-axis for \\0 \leq x \leq 4, giving your answer to three decimal places and using:
The left-endpoint approximation with 4 rectangles.
The right-endpoint approximation with 4 rectangles.
Technology and the left-endpoint approximation with 50 rectangles.
Use integration to find the exact area between the function and the x-axis for 0 \leq x \leq 4.
Calculate the percentage error of the approximation in (a) part (iii) compared to the exact area. Round your answer to the nearest percent.
The following graph shows the function y = \sqrt{x + 1}:
Find an estimate for the area between the function and the x-axis for \\2 \leq x \leq 6, giving your answer to three decimal places and using:
The left-endpoint approximation with 4 rectangles.
The right-endpoint approximation with 4 rectangles.
Technology and the right-endpoint approximation with 40 rectangles.
Use integration to find the area between the function and the x-axis for 2 \leq x \leq 6. Round your answer to four decimal places.
Calculate the percentage error of the approximation in (a) part (iii) compared to area calculated by integration in part (b). Round your answer to two decimal places.
The following graph shows the function f \left( x \right) = \cos x:
Find an estimate for the area between the function and the x-axis for \\0 \leq x \leq \dfrac{\pi}{2}, giving your answer to three decimal places and using left-endpoint approximation with:
3 rectangles.
5 rectangles.
Technology and 50 rectangles.
Use integration to find the exact area between the function and the x-axis for 0 \leq x \leq \dfrac{\pi}{2}.
Calculate the percentage error of the approximation in (a) part (iii) compared compared to the exact area. Round your answer to the nearest percent.
Consider the function f\left(x\right) = xe^{-x} and the given graph of y=f\left(x\right):
Find f'\left(x\right).
Hence, show that: \\ \int xe^{-x}=-xe^{-x}-e^{-x}+c.
Calculate an overestimate for the area bounded by f\left(x\right) and the x-axis for \\1 \leq x \leq 5, giving your answer to three decimal places and using:
2 rectangles of equal width.
4 rectangles of equal width.
Find the exact area bounded by f\left(x\right) and the x-axis over 1 \leq x \leq 5, using part (b).
Calculate the percentage error for the two estimates given in part (c). Round your answer to two decimal places.
Consider the function f\left(x\right) =e^{x}+10-6e^{-x} and the given graph of y=f\left(x\right):
Calculate an overestimate for the area bounded by f\left(x\right) and the x-axis for \\1 \leq x \leq 3 using 4 rectangles of equal width. Give your answer to four decimal places.
Calculate an underestimate for the area bounded by f\left(x\right) and the x-axis for \\1 \leq x \leq 3 using 4 rectangles of equal width. Give your answer to four decimal places.
Given f''\left(x\right)>0 for x>0.9, which estimate will be closer to the exact area bounded by f\left(x\right) and the x-axis for \\1 \leq x \leq 3. Explain your answer.
Consider the function f\left(x\right) =\dfrac{5x}{\sqrt{x^2+1}} and the given graph of y=f\left(x\right):
Calculate an overestimate for the area bounded by f\left(x\right) and the x-axis for \\0 \leq x \leq 2 using 2 rectangles of equal width. Give your answer to three decimal places.
Calculate an overestimate for the area bounded by f\left(x\right) and the x-axis for \\2 \leq x \leq 4 using 2 rectangles of equal width. Give your answer to three decimal places.
Which of the estimates calculated in (a) and (b) will be closer to the exact area being estimated? Explain your answer.
Would an underestimate or overestimate using 4 rectangles of equal width provide a closer estimate to the exact area bounded by f\left(x\right) and the x-axis for 0 \leq x \leq 4. Explain your answer.
The following graph shows the function y = x^{2} - 4:
Find the area A.
Find the area B.
Find the ratio of the areas A:B.
The following graph shows the function y = 3x^{2}:
Let A define the area under the curve over the interval \left[0, k \right] and B define the area under the curve over the interval \left[k, 2k \right]. Find the ratio of the areas A:B.
The following graph shows the function y = \cos x:
Find k such that ratio A:B is 1:1.
Consider the function f \left( x \right) = e^{x}. Let A define the area under the curve over the interval \left[0, k \right] and B define the area under the curve over the interval \left[0, 2k \right]. Find the relationship between A and B.
The curve y = 8 \pi - 2 x + \sin x is shown passing through the point P \left( \pi, 6 \pi \right). A straight line joins the origin to point P.
Find the value of \int_{0}^{\pi} \left( 8 \pi - 2 x + \sin x\right) dx.
Find the value of A.
Find the value of B.
Determine the ratio A:B in the form \\1:k. Round k to two decimal places.
Consider the function f \left( x \right) = 6 a x - 6 x^{2}, for a \geq 0.
For a = 1, sketch the function.
Find the area bound by the curve and the x-axis.
Use your calculator to complete the table of the area bound by the graph and the x-axis for different values of a.
| a | 1 | 2 | 3 | 5 | 10 |
|---|---|---|---|---|---|
| \text{Area }(\text{units})^2 |
Consider the function f \left( x \right) = k x^{n} \left(1 - x\right) on the interval \left[ 0 ,1 \right], where n is a positive integer.
For n = 1, find the value of k if the area under the curve on the interval \left[0, 1 \right] is 1\text{ unit}^2.
Using technology, find the area under the curve y = x^{2} \left(1 - x\right) on the interval \left[ 0, 1 \right].
Hence, find the value of k that would create an area of 1\text{ unit}^2 under the curve of f \left( x \right) over the interval \left[ 0, 1 \right], when n = 2.
Complete the following table showing the value of k required for different values of n, if the area on the interval \left[0, 1\right] is 1\text{ unit}^2:
| n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| k |
Consider the curve y = f_k \left( x \right) where f_k \left( x \right) = \cos \left( kx \right) and k is a positive integer.
The domain of the function f_k \left( x \right) is dependent on k which is 0 \leq x \leq \dfrac{\pi}{2 k}.
The given graph is of f_1 \left( x \right), where \\k = 1, thus f_1 \left( x \right) = \cos \left( 1 x \right) and the domain is 0 \leq x \leq \dfrac{\pi}{2}.
Find the area between y = f_1 \left( x \right) and the x-axis over the domain of the function.
Consider y = f_k \left( x \right) for k = 2:
State the function forf_2 \left( x \right).
State the domain of f_2 \left( x \right).
Find the area between y = f_2 \left( x \right) and the x-axis over the domain of the function.
Find the area under f_3 \left(x\right) over its domain.