Volume of y = 1 - x^2 from x = 0 to x = 1 turned around the y-axis Step by Step

Step by step

The question

Question: find the volume when the region under y = 1 − x² from x = 0 to x = 1 turns around the y-axis.

y = 1 − x²

1 Draw and turn it

Draw the curve first.

The region under the curve 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 height of the curve: h = 1 − x².

Its thickness is a tiny dx.

h = 1 − x²

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 1: that is an integral.

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

V = 2π∫ from 0 to 1 of x(1 − x²) dx

Careful!

The radius is not the height of the curve.

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

Careful: not radius = 1 − x² ✗

4 Multiply

Multiply x into the bracket: x times 1 − x².

V = 2π∫ from 0 to 1 of (x − x³) dx

x times every part.

x(1 − x²) = x − x³

5 The antiderivative

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

V = 2π[ x²2 − x⁴4 ] from 0 to 1

6 Top minus bottom

Put in 1, then take away the same with 0.

= 2π[(1²2 − 1⁴4) − (0²2 − 0⁴4)]

= 2π[14 − 0]

= 2π × 14

2π times 1/4: the volume is π over 2 cubic units.

That is about 1.57.

V = π2 ≈ 1.57

7 Check

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

That gives about 1.57, the same as our answer.

Right!

1000 thin shells: ≈ 1.57

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