Sketch the region enclosed by the given curves. Decide whether to integrate with

Question

1. Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle.

y = x2 - 4x, y = 2x + 7 Find the area of the region

1. Sketch the region enclosed by the given curves.

y = |4x|, y = x2 -5 Find the area of the region

1. Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

y = 4- 4x2, y = 0 Find the area of the region

1. Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
2. y =(1/25)x2, x = 5, y = 0; about the y-axis

Find the area of the region y = x2 - 4x, y = 2x + 7 Find the area of the region

1. Sketch the region enclosed by the given curves.

y = |4x|, y = x2 -5 Find the area of the region

1. Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

y = 4- 4x2, y = 0 Find the area of the region

1. Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
2. y =(1/25)x2, x = 5, y = 0; about the y-axis

Find the area of the region y = x2 - 4x, y = 2x + 7 Find the area of the region

1. Sketch the region enclosed by the given curves.

y = |4x|, y = x2 -5 Find the area of the region

1. Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

y = 4- 4x2, y = 0 Find the area of the region

1. Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
2. y =(1/25)x2, x = 5, y = 0; about the y-axis

Find the area of the region y = |4x|, y = x2 -5 Find the area of the region

1. Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

y = 4- 4x2, y = 0 Find the area of the region

1. Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
2. y =(1/25)x2, x = 5, y = 0; about the y-axis

Find the area of the region y = |4x|, y = x2 -5 Find the area of the region

1. Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

y = 4- 4x2, y = 0 Find the area of the region

1. Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
2. y =(1/25)x2, x = 5, y = 0; about the y-axis

Find the area of the region y = 4- 4x2, y = 0 Find the area of the region

1. Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
2. y =(1/25)x2, x = 5, y = 0; about the y-axis

Find the area of the region y = 4- 4x2, y = 0 Find the area of the region

1. Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
2. y =(1/25)x2, x = 5, y = 0; about the y-axis

Find the area of the region Find the area of the region Find the area of the region

Explanation / Answer

o find the points of intersection of the curves, we solve the equations of the curves simultaneously.
Therefore
x^2 - 4 = 2x - x^2
x^2 - 4 - 2x + x^2 = 0
2x^2 - 2x - 4 = 0
Divide by 2
x^2 - x - 2 = 0
(x+1)(x-2) = 0
x = -1, 2

The area A = |integral (x^2 - 4) - (2x - x^2) dx from -1 to 2|
= |integral x^2 - 4 - 2x + x^2 dx from -1 to 2|
= |integral 2x^2 - 2x - 4 dx from -1 to 2|
= | 2x^3/3 - 2x^2/2 - 4x from -1 to 2 |
= | 2x^3/3 - x^2 - 4x from -1 to 2 |
= | (2 * 2^3/3 - 2^2 - 4*2) - [2*(-1)^3/3 - (-1)^2 - 4(-1)] |
= |(16/3 - 4 - 8) - (-2/3 - 1 + 4)|
= |16/3 - 12 + 2/3 + 1 - 4|
= |16/3 + 2/3 - 12 + 1 - 4|
= |18/3 - 15|
= |6 - 15|
= |-9|
= 9

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