Practice question · Multiple choice
The Fundamental Theorem connects the derivative and the integral, which look like unrelated operations. Why should differentiation and area-finding be inverse to each other?
Hints
- Let A(x) be the area accumulated from a to x. Nudge x forward slightly - how much area is added?
- Ask what the rate of change of accumulated area is.
Show the answer
C. Because the area accumulated up to x grows at the height of the curve.
Why
Push x forward by h and the extra area is a sliver of width h and height about f(x), so A′ = f, the area function is an antiderivative, and Part 2 follows since any other differs by a constant. Both objects are defined independently, one as a limit of Riemann sums and the other by differentiation; that they coincide is a theorem, and it is what made integrals computable.
Practise The Fundamental Theorem of Calculus
The app has 5 more questions on this lesson, and keeps your place in the course. Computer Science I is free to start.
More questions on The Fundamental Theorem of Calculus
- Order the steps of evaluating a definite integral using the Fundamental Theorem.
- Euler's method has local error O(h²) per step and global error O(h). Where does the lost order go?
- Complete the statement of the first part of the theorem.
- Define F(x) as the definite integral of t^2 from t = 0 up to t = x. Using the first part of the theorem,…
- Select every statement that follows from the Fundamental Theorem of Calculus.