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Definite Integral Practice Problems

Sketch the graph, split the region into simple shapes, and subtract any area below the x-axis. For the property problems, combine the given integrals instead of integrating. Type exact answers like 2pi or decimals rounded to three places.

  1. 1.

    Find .

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    The region is a rectangle of height 4 and width 3, so .

  2. 2.

    Find using geometry.

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    The region is a trapezoid with parallel sides and and width 4: .

  3. 3.

    Find .

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    The interval has zero width, so for any .

  4. 4.

    Find .

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    The graph of is the upper half of the circle , radius 2. The integral is the area of the semicircle: .

  5. 5.

    Find .

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    From 0 to 3 the line is below the axis: a triangle with base 3 and height 3, area 4.5. From 3 to 5 it's above: base 2, height 2, area 2. The net area is .

  6. 6.

    Find the total area between and the x-axis from to .

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    Total area counts both triangles as positive: .

  7. 7.

    Given and , find .

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    By additivity, , so .

  8. 8.

    Given , find .

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    Reversing the limits changes the sign: .

  9. 9.

    Given and , find .

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    Split and pull out the constants: .

  10. 10.

    Given , find .

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    .

  11. 11.

    Find without computing any sums. (Hint: think about symmetry.)

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    is an odd function, so the region from to 0 is the mirror image of the region from 0 to 1, below the axis instead of above. The two signed areas cancel, and the integral is 0.

  12. 12.

    The limit is a definite integral . Find its value. (Use .)

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    and , so and . The limit is .
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Definite Integral Practice Problems

These problems follow the definite integral lesson. Each one asks for the value of a definite integral from its graph, from given integrals using the properties, or from a limit of Riemann sums.

The method

  1. Sketch on and mark where it crosses the x-axis.
  2. Find the area of each piece with geometry.
  3. Add the pieces above the axis and subtract the pieces below.
  4. For property problems, rewrite the integral in terms of the ones you’re given.

Common mistakes to watch for

  • Adding . A definite integral is a number.
  • Mixing up net and total area. Net area subtracts the parts below the axis; total area adds them.
  • Forgetting the sign when the limits are reversed.

Frequently Asked Questions

How do I type answers with π?+

Type pi, as in 2pi or 2*pi, or give a decimal rounded to three places, such as 6.283.

Do I need to add + C?+

No. A definite integral is a number, so none of these answers has a constant of integration.

More practice problems

Riemann Sums Practice Problems Antiderivatives Practice Problems Limits at Infinity Practice Problems