What is the value of ∫ from 0 to 2 of (3x^2 + 1) dx? Step by Step

Step by step

The question

Question: find the integral from 0 to 2 of 3x² + 1.

∫ from 0 to 2 of (3x² + 1) dx

An integral from a to b gives one number.

There are four choices.

First we solve it, the normal way.

Then we pick.

1 The antiderivative

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

F(x) = 3 × x³3 + x

Now we have F(x) = x³ + x.

= x³ + x

No + C here: in F(b) minus F(a), the two Cs cancel.

no + C here: it cancels

2 Top minus bottom

The rule: put in the top number, then take away the bottom number.

∫ from a to b of f(x) dx = F(b) − F(a)

Write F(x) with 0 and 2 on the bar.

= (x³ + x) from 0 to 2

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

= (2³ + 2) − (0³ + 0)

First the powers and the times.

= (8 + 2) − (0 + 0)

Then each bracket: the first is 10, the second is 0.

= 10 − 0

Careful!

Top first, then take away the bottom.

The other way gives the wrong sign.

Careful: not F(0) − F(2) ✗

So the integral is 10.

= 10

3 Check

Let's check F: its derivative is 3 x² + 1, the function we started with.

Right!

F′(x) = 3x² + 1

4 Pick the choice

Our answer is 10.

That is choice B.

B) 10

5 Why not the others

A is 8.

Wrong: lost the + 1, which gives x.

A) 8 ✗ lost the + 1, which gives x

C is 13.

Wrong: put 2 into 3x² + 1: put it into the antiderivative.

C) 13 ✗ put 2 into 3x^2 + 1: put it into the antiderivative

D is 12.

Wrong: differentiated and put in 2.

D) 12 ✗ differentiated and put in 2

Let's look back.

First: the antiderivative.

Then: top minus bottom.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer