Caacl 2 need help answer?

Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

y = x,  y = 0,  x = 4,  x = 7;

 about x = 1

Answer 1

Using shell method volume V = 2π∫₄⁷(x - 1)xdx = 2π∫₄⁷(x² - x)dx = 2π(x³/3 - x²/2)₄⁷ = 2π[(7³/3 - 7²/2) -

(4³/3 - 4²/2)] = 2π(343/3 - 49/2 - 64/3 + 8) = 153π

Answer 2

Aisha

Draw a picture of this.

Basically the solid can be split two parts. The part from y = 0 to y = 4. This consists of a cylinder, radius 6 (that is the distance from x=1 to x=7) minus a cylinder radius 3 (distance from x=1 to x=4).

As the volume of a cylinder is πr²h, and the height will be 4, this volume is 4π (6² - 3²) (*)

The second part of the volume will be from y=4 to y=7.

For a given y, the radius of the outer part of this solid from x=1 will be 6 (fixed)

when y=4 the radius of the inner part is 3 (from x=1)

when y = 5 the radius is 4

when y=5 the radius is 5

so the radius will be 3 + (y-4) = y-1

So for a fixed y the volume of the slice thickness dy will be π ( 6 ² - (y-1)² ) dy

So you need to integrate this from y = 4 to y = 7, and add the result to the above result (*)

Mike