Evaluate an Antiderivative at Both Bounds
For a continuous integrand, a suitable antiderivative can evaluate a definite integral by subtracting its lower-bound value from its upper-bound value.
About 25 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: Evaluate the definite integral of 2x from 0 to 2. Candidate answers: (A) 3 (B) 4 (C) 5 (D) 8.
- Independent check inputs: Evaluate the definite integral of 2x from 0 to 3. Candidate answers: (A) 10 (B) 18 (C) 8 (D) 9.
Learn the idea
Integration connects signed accumulation with limiting sums. Rectangle models approximate that accumulation, antiderivatives evaluate suitable definite integrals, and the Fundamental Theorem links an accumulation function’s derivative to its continuous integrand. For a continuous integrand, a suitable antiderivative can evaluate a definite integral by subtracting its lower-bound value from its upper-bound value. For 2x, x² is an antiderivative.
Here is the method on the supplied teaching case. An antiderivative of 2x is x². Evaluate at the bounds and subtract: 2²−0²=4. The additive constant cancels in a definite integral.
Upper minus lower is the required order for the stated orientation. An additive antiderivative constant cancels; a definite integral yields one accumulation value rather than a family of functions.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: Evaluate the definite integral of 2x from 0 to 2. Candidate answers: (A) 3 (B) 4 (C) 5 (D) 8. Solution: An antiderivative of 2x is x². Evaluate at the bounds and subtract: 2²−0²=4. The additive constant cancels in a definite integral. The final answer is 4. Upper minus lower is the required order for the stated orientation. An additive antiderivative constant cancels; a definite integral yields one accumulation value rather than a family of functions.
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Helpful tips
- For a continuous integrand, a suitable antiderivative can evaluate a definite integral by subtracting its lower-bound value from its upper-bound value.
- Upper minus lower is the required order for the stated orientation.
See the answer and explanation
9
An antiderivative of 2x is x². Evaluate at the bounds and subtract: 3²−0²=9. The additive constant cancels in a definite integral.
Take it further
Changed-input task: Evaluate the definite integral of 2x from 0 to 9. Candidate answers: (A) 162 (B) 80 (C) 81 (D) 82. Show the mathematical reason, not only the selected answer. Expected result: 81. An antiderivative of 2x is x². Evaluate at the bounds and subtract: 9²−0²=81. The additive constant cancels in a definite integral.
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