Definite Integrals


In Exercises 5–8, express each limit as a definite integral. Then evaluate the integral to find the value of the limit. In each case, P is a partition of the given interval, and the numbers cₖ are chosen from the subintervals of P.


     n
 lim  ∑ (cos(cₖ/2)) ∆xₖ, where P is a partition of [-π, 0]
 ∥P∥→0  k = 1

In Exercises 75–78, find dy/dx.

y = ∫(from x to 1) (6/(3 + t^4))dt

If ∫²₋₂ 3ƒ(x) dx = 12, ∫⁵₋₂ ƒ(x) dx = 6, and ∫⁵₋₂ g(x) dx = 2, find the value of each of the following.

d. ∫⁵₋₂ (-πg(x)) dx

Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.

√x + √y = 1, x = 0, y = 0

If ∫₀² ƒ(x) dx = π, ∫₀² 7g(x) dx = 7, and ∫₀¹ g(x) dx = 2, find the value of each of the following.

b. ∫₁² g(x) dx

In Exercises 11–14, find the total area of the region between the graph of f and the x-axis.

ƒ(x) = 5 - 5x²/³, -1 ≤ x ≤ 8

           10           10

Suppose that Σ aₖ = -2 and Σ bₖ = 25. Find the value of

           k = 1          k = 1

  10

a. Σ aₖ/4

  k = 1