Study guide: 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.
Ideas to explain in your own words
- A rectangle sum multiplies each supplied height by its interval width and adds the contributions. Equal widths can be factored out, but the result remains a chosen approximation to a specified accumulation. A rectangle sum is not automatically exact or equipped with an error bound.
- 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. Upper minus lower is the required order for the stated orientation.
- A definite integral counts above-axis regions positively and below-axis regions negatively. Net signed area can therefore be smaller than the sum of their geometric magnitudes. A below-axis region’s geometric area is positive, but its integral contribution is negative.
- 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. This is a Fundamental Theorem relationship. Continuity is satisfied by the supplied polynomial integrand.
Useful vocabulary
- definite integral
- Signed accumulation over an oriented interval.
- antiderivative
- A function whose derivative is the integrand.
- integrand
- The function being integrated.
- rectangle sum
- An approximation built by adding height-times-width contributions.
Watch your reasoning
- A rectangle sum is not automatically exact or equipped with an error bound. The supplied heights and widths define this sum; changing the sampling positions can change the approximation for a nonconstant function.
- 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.
- A below-axis region’s geometric area is positive, but its integral contribution is negative. Determine whether the question asks for net accumulation or total area before choosing subtraction or addition.
A worked review example
Complete supplied guide examples: Add Supplied Rectangle Contributions — Approximate accumulation over three equal intervals of width 2 using supplied heights 1,2,3. What is the rectangle sum? Candidate answers: (A) 11 (B) 12 (C) 13 (D) 24. Each rectangle contributes height×width. Sum 2×1+2×2+2×3=12. Without the actual function, these supplied heights define an approximation, not a certified error bound. Evaluate an Antiderivative at Both Bounds — Evaluate the definite integral of 2x from 0 to 2. Candidate answers: (A) 3 (B) 4 (C) 5 (D) 8. An antiderivative of 2x is x². Evaluate at the bounds and subtract: 2²−0²=4. The additive constant cancels in a definite integral. Separate Net Accumulation from Total Geometric Area — A supplied graph has geometric area 6 above the x-axis and 4 below, over one interval. Find its definite integral. Candidate answers: (A) 1 (B) 2 (C) 3 (D) 4. The definite integral accumulates signed area: 6−4=2. The total geometric area would add them to 10, which answers a different question. Differentiate an Accumulation Function — Let F(x)=integral from 0 to x of (t²+1) dt. Find F′(2). Candidate answers: (A) 10 (B) 4 (C) 5 (D) 6. The integrand is continuous, so the Fundamental Theorem gives F′(x)=x²+1. At 2, this equals 5. No integration constant or extra power is needed.
Make a practice plan
- Study the four methods in the listed order; attempt each independent check before opening its explanation.
- Complete each lesson’s three changed-input questions and record the operation or reasoning step behind any correction.
- Take the four-question unit review, then solve the transfer task with written justifications.
- After correcting errors, attempt the separate unit exam and revisit the linked lesson for any unresolved method.