Calculus Volume 1 · free preview
The Fundamental Theorem of Calculus
The fundamental theorem connects differentiation and integration, the definite integral is computed by antiderivative evaluation at the endpoints, and its corollary makes the integral a function whose derivative recovers the integrand.
1. The theorem and its meaning
The theorem and its meaning
2. Part one in action
Part one in action
3. Part two in action
Part two in action
4. Why the theorem is fundamental
Why the theorem is fundamental
5. The theorem's two parts
Which pair of statements correctly distinguishes the two parts of the fundamental theorem of calculus?
6. Area under the parabola
Using the fundamental theorem, what is the integral from 0 to 1 of x^2 dx?
7. Sine arch area
What is the integral from 0 to pi of sin x dx, the area under one arch of the sine curve?
8. Accumulation function derivative
For A(x) = integral from 0 to x of e^(t^2) dt, which expression equals A prime of x?
9. Chain rule upper limit
For A(x) = integral from 0 to x^2 of sin t dt, what is A prime of x?
10. Lower limit variation
For B(x) = integral from x to 5 of t^2 dt, which expression equals B prime of x?
11. Endpoint evaluation notation## practice-fill Endpoint evaluation notation
The expression F(b) - F(a) is commonly written with F and a vertical bar. What symbol sits between F and the limits? Answer with a single character.
12. Area under x cubed
What is the integral from 0 to 2 of x^3 dx? Answer with a number.
13. Reversed accumulation derivative
A(x) = integral from x to 3 of cos t dt. Reversing limits, A prime of x equals what function of x? Answer with a function like -cos x.
14. Splitting at a zero
To integrate |x| from -1 to 1, the interval is split at 0 into two pieces. How many pieces result? Answer with a number.
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