The region between y = x and y = x^2 is turned around the x-axis… Step by Step

Step by step

The question

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

y = x

We turn a flat region around the x-axis, and slice the solid it makes.

There are four choices.

First we solve it, the normal way.

Then we pick.

y = x²

1 Draw and turn it

Draw the curves first.

2 Where they meet

The region ends where the curves meet: where the two y's are equal.

x = x²

Move everything to the side with the x², so the other side is 0.

Then we can factor.

x² − x = 0

Every term has an x in it: take it out.

x(x − 1) = 0

A product is 0 when one of its parts is 0.

So x = 0, or x = 1.

x = 0 or x − 1 = 0 → x = 1

The region between them, from x = 0 to x = 1, is shaded.

3 One slice

Cut the solid into thin slices.

Each slice is a washer: a flat disk with a hole.

The outside radius R is the curve further from the axis.

R = x

Which curve is further?

Both are above the axis here, so the bigger one is further.

Test a number between: x is bigger there.

So R = x.

test x = 0.5: 0.5 > 0.25

The inside radius, the hole, is the other one: r = x².

r = x²

A washer is the big circle minus the hole: π R² minus π r², times its thickness dx.

slice = π(R² − r²) dx

4 Write the integral

Add up all the slices from 0 to 1: that is an integral.

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

V = π∫ from 0 to 1 of (x² − (x²)²) dx

Careful!

Not R minus r, squared!

The hole is a whole circle: take away π r squared.

Careful: not π(R − r)² ✗

5 Square it

Square the radius: square the number and the power: x².

x² = x²

And the hole: x⁴.

(x²)² = x⁴

Take the hole away from the big circle.

V = π∫ from 0 to 1 of (x² − x⁴) dx

6 The antiderivative

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

V = π[ x³3 − x⁵5 ] from 0 to 1

7 Top minus bottom

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

= π[(1³3 − 1⁵5) − (0³3 − 0⁵5)]

= π[215 − 0]

So the volume is 2π over 15 cubic units.

That is about 0.42.

V = 2π15 ≈ 0.42

8 Check

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

That gives about 0.42, the same as our answer.

Right!

1000 thin washers: ≈ 0.42

9 Pick the choice

Our answer is V = 2π over 15 ≈ 0.42.

That is choice A.

A) 0.42

10 Why not the others

B is 0.10.

Wrong: squared the gap x − x²: take the two squares away.

B) 0.10 ✗ squared the gap x − x^2: take the two squares away

C is 1.05.

Wrong: only the solid of y = x.

C) 1.05 ✗ only the solid of y = x

D is 0.13.

Wrong: left out π.

D) 0.13 ✗ left out π

Let's look back.

First: draw and turn it.

Then: where they meet.

Then: one slice.

Then: write the integral.

Then: square it.

Then: the antiderivative.

Then: top minus bottom.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer