Math 2414 Section 7 8 Improper Integrals Infinite Intervals Definitions Improper Integrals over Infinite Intervals 1 b f x dx lim f x dx b a a a If f is continuous on then provided the limit exists b 2 b then If f is continuous on b f x dx lim f x dx c a f x dx lim a provided the limit exists b f x dx lim f x dx a a 3 If f is continuous on then both limits exist and c is any real number such that a c b b c provided In each case if the limit exists the improper integral is said to converge if it does not exist the improper integral is said to diverge e 3x Evaluate 0 dx 1 The limit exists so the integral converges and the region under the curve has a finite area of 3 units2 Evaluate 1 1 x 0 2 dx 1 Let R be the region bounded by the graph of y x and the x axis for x 1 What is the volume of the solid that is generated when R is revolved about the x axis 1 Let R be the region bounded by the graph of y x and the x axis for x 1 What is the volume of the solid that is generated when R is revolved about the y axis Unbounded Integrands Improper integrals also occur when the integrand becomes infinite somewhere on the interval of f x 1 x integration Consider the function What happens to the area under the curve as c 0 Definitions Improper Integrals with an Unbounded Integrand b 1 a b Suppose f is continuous on lim f x with Then x a b f x dx lim f x dx a c a c provided the limit exists b 2 a b Suppose f is continuous on lim f x with Then x b c f x dx lim f x dx a c b a provided the limit exists 3 Suppose f is continuous on b p a b except at the interior point p where f is unbounded b f x dx f x dx f x dx a Then a of the equation exist p provided the improper integrals on the right side In each case if the limit exists the improper integral is said to converge if it does not exist the improper integral is said to diverge Find the area of the region R between the graph of 3 3 if it exists 2 Determine whether the improper integral x 1 x 2 1 f x 1 9 x 2 and the x axis on the interval dx converges or diverges 5p 4 Find cot x dx p 2 10 Evaluate dx x 2 1 13