Each problem is worth ten points. Show all of your work. Evaluate f (3x^2 + 1) (

Question

Each problem is worth ten points. Show all of your work. Evaluate f (3x^2 + 1) (x^3 + x + 1)^6 dx Find the volume of the surface obtained by revolving the bounded by x = 0, x = 1, y = 0, y = x^2 about the x-axis. Find the length of the curve y = 5x from x = 0 to x = 1, Please use the are length formula. Determine if the series sigma^infinity_n = 1 1/2n^2 - 1 converges. Determine if the series sigma^infinity_n = 1 1/4n + 1 converges. Determine if the series sigma^infinity_n = 1 2n/n! converges. Determine if the series sigma^infinity_n = 1 8n^5 + 6n/2n^5 + 5n^2 + 6 converges. Determine if the series 1/3 - 1/27 + 1/81 - 1/243 + ... converges and if it converges, what does it converge to? Find all real numbers x such that sigma^infinity_n = 0 x^n/4^n converges. Up to degree 2, write the Taylor series for the function f(x) = (1 + x)^-5 centered at zero.

Explanation / Answer

(1) (3x^2 + 1)(x^3 + x + 1)^6 dx

Let x^3 + x + 1 = u then (3x^2 + 1)dx = du

u^6 du

= (1/7)u^7 + C

= (1/7)(x^3 + x + 1)^7 + C

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