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Grade 12 · unit review

Review: Introductory Calculus: Integrals and Accumulation

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.

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Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.

Unit review questions
1. Approximate accumulation over three equal intervals of width 7 using supplied heights 1,2,3. What is the rectangle sum?
Need a hint?
  • A rectangle sum multiplies each supplied height by its interval width and adds the contributions.
  • A rectangle sum is not automatically exact or equipped with an error bound.
2. Evaluate the definite integral of 2x from 0 to 7.
Need a hint?
  • 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.
3. A supplied graph has geometric area 21 above the x-axis and 14 below, over one interval. Find its definite integral.
Need a hint?
  • A definite integral counts above-axis regions positively and below-axis regions negatively.
  • A below-axis region’s geometric area is positive, but its integral contribution is negative.
4. Let F(x)=integral from 0 to x of (t²+1) dt. Find F′(7).
Need a hint?
  • If a continuous function is accumulated from a fixed lower bound to variable upper input x, the derivative of that accumulation equals the integrand evaluated at x.
  • Continuity is satisfied by the supplied polynomial integrand.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. Approximate accumulation over three equal intervals of width 9 using supplied heights 1,2,3. What is the rectangle sum? Candidate answers: (A) 108 (B) 53 (C) 54 (D) 55. 2. Evaluate the definite integral of 2x from 0 to 9. Candidate answers: (A) 162 (B) 80 (C) 81 (D) 82. 3. A supplied graph has geometric area 27 above the x-axis and 18 below, over one interval. Find its definite integral. Candidate answers: (A) 10 (B) 18 (C) 8 (D) 9. 4. Let F(x)=integral from 0 to x of (t²+1) dt. Find F′(9). Candidate answers: (A) 164 (B) 81 (C) 82 (D) 83.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. 54. Each rectangle contributes height×width. Sum 9×1+9×2+9×3=54. Without the actual function, these supplied heights define an approximation, not a certified error bound. 2. 81. An antiderivative of 2x is x². Evaluate at the bounds and subtract: 9²−0²=81. The additive constant cancels in a definite integral. 3. 9. The definite integral accumulates signed area: 27−18=9. The total geometric area would add them to 45, which answers a different question. 4. 82. The integrand is continuous, so the Fundamental Theorem gives F′(x)=x²+1. At 9, this equals 82. No integration constant or extra power is needed.

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