# 2.5. Univariate Calculus: Integrals

### Learning Objectives

You should be able to

• Understand what integration is, and how it relates to areas under curves and derivatives
• Calculate approximations of integrals from Riemann sums
• Calculate indefinite integrals of simple functions, such as polynomials, exponential functions and fraction of the forms $$\frac{a}{bx+c}$$
• Use the fundamental theorem of calculus to compute definite integrals of simple functions such as those above

### Get Started: What to do next

If you can pass the diagnostic test, then we believe you are ready to move on from this topic.
Our recommendation is to:
1. Test your ability by trying the diagnostic quiz. You can reattempt it as many times as you want and can leave it part-way through.
2. Identify which topics need more work. Make a note of any areas that you are unable to complete in the diagnostic quiz, or areas that you don't feel comfortable with.
3. Watch the tutorials for these topics, you can find these below. Then try some practice questions from the mini quiz for that specific topic. If you have questions, please ask on the forum.
4. Re-try the diagnostic quiz. And repeat the above steps as necessary until you can pass the diagnostic test. A pass grade is 80%.

Optionally, also see if you can apply these skills to an economic application or look at the additional resources and advanced quizzes at the bottom of the page,

### Tutorials: How to guides

#### 2.5.1. What is an integral?

See how to think about integration in terms of areas under curves, limits of Riemann sums and as an inverse to differentiation.

Please note that we didn't have time to make a video on this topic, so instead, the video below is taken from an excellent YouTube channel 3Blue1Brown.

#### 2.5.2. Computing indefinite integrals as antiderivatives

$$\int f(x) \mathrm{d}x = F(x) + C \iff \frac{\mathrm{d}F}{\mathrm{d}x}(x) = f(x)$$

#### 2.5.3. Definite integrals and the fundamental theorem of calculus

$$F(t) = \int_c^t f(x) \mathrm{d}x$$,      $$F'(x) = f(x)$$,      $$\int_a^b f(x)\mathrm{d}x = F(b)-F(a)$$

### Economic Application: Consumer surplus

In analyzing the welfare implications of different market structures (for example, whether monopolies are good or bad for consumers compared to competitive markets) economists make use of concepts such as producer surplus, consumer surplus and total welfare. In standard models of the market, the value of these surplus can be represented graphically as certain areas in a supply and demand diagram. This application introduces the concept of consumer surplus, and show how consumer surplus in a market can be calculated as definite integrals.

#### Economic Application Exercise

The following quiz allows you to test your understanding of consumer surplus and its calculation through integration.

### Further practice & resources

You can find practice quizzes on each topic in the links above the tutorial videos, or you can take the diagnostic quiz as many times as you like to If you want to test yourself further, then try the advanced quiz linked below.