Worked examples
The question
Question: find the volume when y = √x from x = 0 to x = 4 turns around the x-axis.
y = √x
1 Draw and turn it
Draw the curve first.
The region under the curve, from x = 0 to x = 4, is shaded.
2 One slice
Cut the solid into thin slices.
Each slice is a disk: a very flat cylinder.
Its radius is the height of the curve: r = √x.
Its thickness is a tiny dx.
r = √x
A cylinder is its circle's area times its thickness.
So one slice is π r² dx.
slice = πr2 dx
3 Write the integral
Add up all the slices from 0 to 4: that is an integral.
π is just a number, so it waits in front.
V = π∫ from 0 to 4 of (√x)2 dx
Careful!
Square the radius!
A circle's area is π r squared, not π r.
Careful: not V = π∫ from 0 to 4 of √x dx ✗
4 Square it
Square the radius.
V = π∫ from 0 to 4 of x dx
A square root squared gives back what is inside: (√x)² = x.
(√x)2 = x
5 The antiderivative
The antiderivative: add 1 to each power, divide by the new power.
V = π[ x22 ] from 0 to 4
6 Top minus bottom
Put in 4, then take away the same with 0.
= π[(422) − (022)]
= π[8 − 0]
So the volume is 8π cubic units.
That is about 25.13.
V = 8π ≈ 25.13
7 Check
Let's check with a calculator: it adds up 1000 thin disks.
That gives about 25.13, the same as our answer.
Right!
1000 thin disks: ≈ 25.13
Let's look back.
First: draw and turn it.
Then: one slice.
Then: write the integral.
Then: square it.
Then: the antiderivative.
Then: top minus bottom.
Last: check.
▶ Watch it solved
The question
Question: find the volume when y = −x² + 4 from x = −2 to x = 2 turns around the x-axis.
y = −x2 + 4
1 Draw and turn it
Draw the curve first.
The region under the curve, from x = −2 to x = 2, is shaded.
2 One slice
Cut the solid into thin slices.
Each slice is a disk: a very flat cylinder.
Its radius is the height of the curve: r = −x² + 4.
Its thickness is a tiny dx.
r = −x2 + 4
A cylinder is its circle's area times its thickness.
So one slice is π r² dx.
slice = πr2 dx
3 Write the integral
Add up all the slices from −2 to 2: that is an integral.
π is just a number, so it waits in front.
V = π∫ from −2 to 2 of (−x2 + 4)2 dx
Careful!
Square the radius!
A circle's area is π r squared, not π r.
Careful: not V = π∫ from −2 to 2 of −x2 + 4 dx ✗
4 Square it
Square the radius.
V = π∫ from −2 to 2 of (x4 − 8x2 + 16) dx
A bracket squared is the bracket times itself: multiply every part by every part.
(−x2 + 4)2 = (−x2 + 4)(−x2 + 4)
= x4 − 8x2 + 16
5 The antiderivative
The antiderivative: add 1 to each power, divide by the new power.
V = π[ x55 − 8x33 + 16x ] from −2 to 2
6 Top minus bottom
Put in 2, then take away the same with −2.
= π[(255 − 8(2)33 + 16(2)) − ((−2)55 − 8(−2)33 + 16(−2))]
= π[25615 − (−25615)]
So the volume is 512π over 15 cubic units.
That is about 107.23.
V = 512π15 ≈ 107.23
7 Check
Let's check with a calculator: it adds up 1000 thin disks.
That gives about 107.23, the same as our answer.
Right!
1000 thin disks: ≈ 107.23
Let's look back.
First: draw and turn it.
Then: one slice.
Then: write the integral.
Then: square it.
Then: the antiderivative.
Then: top minus bottom.
Last: check.
Answer V = 512π15 ≈ 107.23
▶ Watch it solved
The question
Question: find the volume when y = −x + 2 from x = 0 to x = 2 turns around the x-axis.
y = −x + 2
1 Draw and turn it
Draw the curve first.
The region under the curve, from x = 0 to x = 2, is shaded.
2 One slice
Cut the solid into thin slices.
Each slice is a disk: a very flat cylinder.
Its radius is the height of the curve: r = −x + 2.
Its thickness is a tiny dx.
r = −x + 2
A cylinder is its circle's area times its thickness.
So one slice is π r² dx.
slice = πr2 dx
3 Write the integral
Add up all the slices from 0 to 2: that is an integral.
π is just a number, so it waits in front.
V = π∫ from 0 to 2 of (−x + 2)2 dx
Careful!
Square the radius!
A circle's area is π r squared, not π r.
Careful: not V = π∫ from 0 to 2 of −x + 2 dx ✗
4 Square it
Square the radius.
V = π∫ from 0 to 2 of (x2 − 4x + 4) dx
A bracket squared is the bracket times itself: multiply every part by every part.
(−x + 2)2 = (−x + 2)(−x + 2)
= x2 − 4x + 4
5 The antiderivative
The antiderivative: add 1 to each power, divide by the new power.
V = π[ x33 − 2x2 + 4x ] from 0 to 2
6 Top minus bottom
Put in 2, then take away the same with 0.
= π[(233 − 2(2)2 + 4(2)) − (033 − 2(0)2 + 4(0))]
= π[83 − 0]
So the volume is 8π over 3 cubic units.
That is about 8.38.
V = 8π3 ≈ 8.38
7 Check
Let's check with a calculator: it adds up 1000 thin disks.
That gives about 8.38, the same as our answer.
Right!
1000 thin disks: ≈ 8.38
Let's look back.
First: draw and turn it.
Then: one slice.
Then: write the integral.
Then: square it.
Then: the antiderivative.
Then: top minus bottom.
Last: check.
▶ Watch it solved
