Area between two curves - along x-axis and y-axis

YOU WILL LEARN ABOUT:

How to determine the area between two curves by integrating with respect to $x$ or $y$ , which is a generalization of the concept of finding the area between a curve and the $x$ -axis $(y = 0)$ .

The area between the continuous curves

$y = f_{1} (x)$ and $y = f_{2} (x)$ and between $x = a$ and $x = b$ is given by $\int_{a}^{b} [f_{1} (x) - f_{2} (x)] d x$ , where $f_{1} (x) = f_{2} (x)$ for all $x$ between $a$ and $b$ .
$x = g_{1} (y)$ and $x = g_{2} (y)$ and between $y = c$ and $y = d$ is given by $\int_{d}^{c} [g_{1} (y) - g_{2} (y)] d y$ , where $g_{1} (y) = g_{2} (y)$ for all $y$ between $c$ and $d$ .

Area calculations are important in many applications. The area of many regions can be determined using integrals by thinking of the regions as lying between two curves. The following are examples of such regions.

The area between the marginal revenue and marginal cost functions, from x = a to
x = b, for a manufacturer represents the increase in profits as the production level increases from a to b units.

Medical professionals use a quantity known as quality-adjusted life years (QALYs) as a measure of health outcomes which incorporates both the quality and quantity of life lived. When considering some sort of medical intervention, the area between two QALY vs. time curves represents the gain in quality-adjusted life years by the intervention.

An environmental contractor hired to clean up an oil spill needs to determine the area of an oil slick, which can be modeled as the region between two curves, to be able to determine the proper amount of dispersants and biodegradation agents to use to clean the spill.

Click on the tabs on the right to watch videos on rotating a region about the x-axis and y-axis.

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Y Axis

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y = √ x

y = x²

0.33333 Difference = 0.03380

Number of rectangles:4
Note:The first rectangle has a height of zero.

0.29953

Integrate with respect to x-axis y-axis

Function 1

Function 2

Function 3

Function 4

1 2 3

3. Using 4 rectangles of equal width and left end points, the approximate area between these two curves is found in which of the following ways?

(0 – 0)0.25 + (√0.25 – (0.25)²)0.25 + (√0.5 – (0.5)²)0.25 +(√0.75 – (0.75)²)0.25

(0 – 0)0.25 + ((0.25)² – √0.25)0.25 + ((0.5)² – √0.5)0.25 +((0.75)² – √0.75)0.25

(√0.25 – ((0.25)²).0.25 + (√0.5 – ((0.5)²)0.25 + (√0.75 – ((0.75)²)0.25 + (1 – 1)0.25

None of the above

2. What is the minimum number of rectangles needed for the approximate area between these two curves to become and stay within 0.01 of the actual area?

1. What is Δy if the interval on the y-axis from 0 to 1 is partitioned into 10 rectangles of equal height?

Δy = 0.01

Δy = 0.1

Δy = 0.25

Δy = 1

1 2 3

3. Using 4 rectangles of equal width and left end points, the approximate area between these two curves is found in which of the following ways?

2. What is the minimum number of rectangles needed for the approximate area between these two curves to become and stay within 0.02 of the actual area?

More than 40

1. How would you set up the integrals to find the area between these two curves, if you were integrating with respect to y?

1 2 3

3. In the simulation with 4 rectangles, the approximate area is computed using left end points. How would the approximate area using right end points compare to the approximate area using left end points?

The approximate area using right end point would be greater

The approximate area using right end points would be less

The approximate area using right end points would be the same

Cannot determine the answer based on the graph and given information

2. What is the minimum number of rectangles needed for the approximate area between these two curves to become and stay within 0.1 of the actual area?

1. What is Δx if the interval from 0 to 2 is partitioned into 8 rectangles of equal width?

Δx = 0.25

Δx = 0.5

Δx = 0.75

Δx = 1

1 2 3

3. What is the minimum number of rectangles needed for the approximate area between these two curves to be within 0.001 of the actual area when integrating with respect to y and using the lower edges of the rectangles to determine the width of each rectangle?

2. If you were integrating with respect to y, what is Δy if the interval from π/2 to π is partitioned into 8 rectangles of equal height?

1. How would you set up the integrals to find the area between these two curves, if you were integrating with respect to x?