Volume between y = x + 2 and y = x^2 from x = 0 to x = 2 turned… Step by Step

Step by step

The question

Question: find the volume when the region between y = x + 2 and y = x² turns around the y-axis.

y = x + 2

y = x²

1 Draw and turn it

Draw the curves first.

The region between them is shaded.

2 One shell

Cut the solid into thin tubes, one inside the other.

Each one is a shell: like a can with no top and no bottom.

A shell at x is x away from the y-axis.

That is its radius.

radius = x

Its height is the top curve minus the bottom one.

h = (x + 2) − (x²)

Which is on top?

Test a number between: x + 2 is bigger there.

test x = 1: 3 > 1

So h = x − x² + 2.

= x − x² + 2

Cut a shell open and lay it flat.

It is a thin slab: 2πx long, the distance around a circle of radius x.

h tall, and dx thick.

shell = 2πx × h × dx

3 Write the integral

Add up all the shells from 0 to 2: that is an integral.

2π is just a number, so it waits in front.

V = 2π∫ from 0 to 2 of x(x − x² + 2) dx

Careful!

The radius is not the height of the curve.

It is the distance to the y-axis: x.

Careful: not radius = x − x² + 2 ✗

4 Multiply

Multiply x into the bracket: x times x − x² + 2.

V = 2π∫ from 0 to 2 of (x² − x³ + 2x) dx

x times every part.

x(x − x² + 2) = x² − x³ + 2x

5 The antiderivative

The antiderivative: add 1 to each power, divide by the new power.

V = 2π[ x³3 − x⁴4 + x² ] from 0 to 2

6 Top minus bottom

Put in 2, then take away the same with 0.

= 2π[(2³3 − 2⁴4 + 2²) − (0³3 − 0⁴4 + 0²)]

= 2π[83 − 0]

= 2π × 83

2π times 8/3: the volume is 16π over 3 cubic units.

That is about 16.76.

V = 16π3 ≈ 16.76

7 Check

Let's check with a calculator: it adds up 1000 thin shells.

That gives about 16.76, the same as our answer.

Right!

1000 thin shells: ≈ 16.76

Let's look back.

First: draw and turn it.

Then: one shell.

Then: write the integral.

Then: multiply.

Then: the antiderivative.

Then: top minus bottom.

Last: check.

Answer