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Mathematics, five units · Ages 17–18

Calculating Definite Integrals Practice

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Designed for Grade 12 calculus students. Find the antiderivative for each expression, evaluate it across the specified limits, and compute the exact value.

  1. Evaluate the definite integral of 2x with respect to x from x = 0 to x = 3.

  2. Evaluate the definite integral of 3x^2 with respect to x from x = 1 to x = 2.

  3. Evaluate the definite integral of (4x + 1) with respect to x from x = 0 to x = 2.

  4. Evaluate the definite integral of x^2 with respect to x from x = 1 to x = 2.

  5. Evaluate the definite integral of e^x with respect to x from x = 0 to x = ln(3).

  6. Evaluate the definite integral of cos(x) with respect to x from x = 0 to x = pi/2.

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Calculating Definite Integrals Practice

Answer key

  1. Evaluate the definite integral of 2x with respect to x from x = 0 to x = 3.

    Antiderivative: x^2. Evaluate: 3^2 - 0^2 = 9 - 0 = 9.

  2. Evaluate the definite integral of 3x^2 with respect to x from x = 1 to x = 2.

    Antiderivative: x^3. Evaluate: 2^3 - 1^3 = 8 - 1 = 7.

  3. Evaluate the definite integral of (4x + 1) with respect to x from x = 0 to x = 2.

    Antiderivative: 2x^2 + x. Evaluate: [2(2)^2 + 2] - [0] = 10 - 0 = 10.

  4. Evaluate the definite integral of x^2 with respect to x from x = 1 to x = 2.

    Antiderivative: (1/3)x^3. Evaluate: (1/3)(8) - (1/3)(1) = 7/3.

  5. Evaluate the definite integral of e^x with respect to x from x = 0 to x = ln(3).

    Antiderivative: e^x. Evaluate: e^(ln(3)) - e^0 = 3 - 1 = 2.

  6. Evaluate the definite integral of cos(x) with respect to x from x = 0 to x = pi/2.

    Antiderivative: sin(x). Evaluate: sin(pi/2) - sin(0) = 1 - 0 = 1.