What is the value of ∫ from 1 to 2 of (4x

Step by step

The question

Question: find the integral from 1 to 2 of 4x − 1.

∫ from 1 to 2 of (4x − 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) = 4 × x²2 − x

Now we have F(x) = 2 x² − x.

= 2x² − 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 1 and 2 on the bar.

= (2x² − x) from 1 to 2

Put in 2 first, then take away the same with 1.

= (2(2)² − 2) − (2(1)² − 1)

First the powers and the times.

= (8 − 2) − (2 − 1)

Then each bracket: the first is 6, the second is 1.

= 6 − 1

Careful!

Top first, then take away the bottom.

The other way gives the wrong sign.

Careful: not F(1) − F(2) ✗

So the integral is 5.

= 5

3 Check

Let's check with geometry.

Under a straight line the shape is a trapezoid: heights 3 and 7, width 1.

Its area is 5.

The same!

Right!

trapezoid: 3 + 72 × 1 = 5

4 Pick the choice

Our answer is 5.

That is choice C.

C) 5

5 Why not the others

A is 7.

Wrong: put 2 into 4x − 1: put it into the antiderivative.

A) 7 ✗ put 2 into 4x − 1: put it into the antiderivative

B is 6.

Wrong: used the top limit only.

B) 6 ✗ used the top limit only

D is 2.

Wrong: 4x gives 2x², not x².

D) 2 ✗ 4x gives 2x^2, not x^2

Let's look back.

First: the antiderivative.

Then: top minus bottom.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer

= 5

What is the value of ∫ from 1 to 2 of (4x - 1) dx?