NZ Level 8 (NZC) Level 3 (NCEA) [In development] Calculate definite integrals
Lesson

Now that we've taken a look at approximating areas under a curve, using geometry to find areas and we also understand the Fundamental Theorem of Calculus, we're ready to launch into calculus involving definite integrals.

Definite Integration

A three-step process:

1. Calculate the antiderivative, $F(x)$F(x)
2. Substitute in the upper and lower limits and subtract: $F(b)-F(a)$F(b)F(a)
3. Evaluate Once you've calculated the antiderivative, we use square brackets around our primitive function and we now place the upper and lower limits on the right bracket. This notation shows in one step the integral and what we're getting ready to substitute into our antiderivative.

Watch how it's done in the two worked examples below.

#### Worked Examples

##### Example 1

Evaluate  ##### Example 2

Evaluate  #### Worked Questions

##### Question 1

Calculate $\int_2^4\left(6x+5\right)dx$42(6x+5)dx.

##### Question 2

Calculate $\int_{-2}^1\left(x^2+4\right)dx$12(x2+4)dx.

##### Question 3

Calculate $\int_6^{11}\sqrt{x-2}dx$116x2dx.

### Outcomes

#### M8-11

Choose and apply a variety of differentiation, integration, and antidifferentiation techniques to functions and relations, using both analytical and numerical methods

#### 91579

Apply integration methods in solving problems