I need Some help 1/1 - x = xn, ex = xn/n!, arctan x = (-1)n x 2n + 1/2n + 1, sin

Question

I need Some help

1/1 - x = xn, ex = xn/n!, arctan x = (-1)n x 2n + 1/2n + 1, sinx = (-1)n x 2n + 1/(2n + 1)!, cosx = (-1)n x2n/(2n)! Find the volume of the solid generated by revolving the region bou y = x2 and y = 2x about the x-axis. Use the shell method to find the volume of the solid generated by the region bounded by y = 2x - x2, y = 0 baout the y-axis. int x2 In x dx sin3theta cos5 theta d theta int x - 1/x3 + 4x2 + 3x dx int rootx2 - 9/x dx Determine whether the series is n/100n + 6 3n/n! Find the interval of convergence of the series (4x - 1)n/n3n Given the parametric curve x = t + sint, y = 1 - cost. Find an equation for the line tangent to the curve at the point specified by t = pi/6. Also, find the value of d2y/dx2 at this point. Given the polar equation for the cardioid r = 1 + Sketch the curve. Find the derivative dy/dx. Find the area of the region enclosed by the cardioid. find the length of the cardioid. 4n + 3 n+1/5n 1+3+9/2!+27/3!+81/4!+243/5!+... (-1)n pi 2n+1/4 2n+1(2n + 1)

Explanation / Answer

Q.12

a)first term is a geometric progression.

sum=(4/5)/(1-(4/5))=4

second term is also an infinite gemotetric progression with first term as 9/5 and ratio as 3/5

so sum=(9/5)/(1-(3/5))=9/2

so total sum=4+(9/2)=17/2

b)we know exp(x)=1+x+(x^2/2!)+(x^3/3!)+...

comparing

we get x=3

so sum=exp(3)=20.08

c)we know sin x=x-(x^3/3)+(x^5/5)-...

comapring,

x=pi/4

so sum=sin(pi/4)=1/sqrt(2)=0.707

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