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  ∑ (2cₖ - 1)⁻¹/² ∆xₖ, where P is a partition of [1, 5]
 ∥P∥→0  k = 1

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

y = 2 sin x, y = sin 2x, 0 ≤ x ≤ π

Evaluate the integrals in Exercises 47–68.

∫₀¹/² x³ (1 + 9x⁴)⁻³/² dx

Area

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

ƒ(x) = x² - 4x + 3, 0 ≤ x ≤ 3

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

y = sin x, y = x, 0 ≤ x ≤ π/4

Evaluate the integrals in Exercises 47–68.

∫₁² 4 dv

v²

Express the solutions of the initial value problems in Exercises 35 and 36 in terms of integrals.

dy/dx = sin x/x , y(5) = -3