How to Evaluate a Definite Integral

Step-by-step solution

1. Find the antiderivative.

   $$\int 2x\,dx = x^2$$

   Why: The power rule for integrals gives x squared.

2. Write the antiderivative with the limits.

   $$\left[x^2\right]_1^3$$

   Why: The limits stay on the bracket until you evaluate.

3. Substitute the top limit, then the bottom limit.

   $$3^2 - 1^2 = 9 - 1$$

   Why: A definite integral is the top value minus the bottom value.

4. Subtract.

   $$9 - 1 = 8$$

   Why: The integral measures the area between the graph and the x-axis.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

1. 1

   $\int_0^2 x\,dx$

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   $2$

2. 2

   $\int_0^3 2x\,dx$

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   $9$

3. 3

   $\int_1^2 3x^2\,dx$

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   $7$

4. 4

   $\int_0^1 4x^3\,dx$

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   $1$

5. 5

   $\int_1^4 2x\,dx$

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   $15$