Calculus

Volume of Revolution

Volume of Revolution - Disc Method

         

Let VV be the volume of the solid obtained by revolving the curve y=x2 y = x^2 from x=2 x = 2 to 33 about the xx-axis. What is 10V?10 V?

Let VV be the volume of the solid obtained by revolving the curve x2+y2100=0 x^2+y^2 -100 =0 from x=0 x = 0 to x=4x=4 about the xx-axis. What is 3V?3 V?

If aa is a positive number such that the volume of the solid obtained by rotating the ellipse x2+ay2=1 {x}^2 + a{y}^2 = 1 around the xx-axis is 457π, \frac{4}{57}\pi , what is a?a?

The yellow-colored region in the above diagram is bounded by y=18sinxa (0<a <18),x=0,x=π,y=0.\begin{array}{c}&y= 18\sin x - a \ (0 < a \ < 18), &x=0, &x=\pi, &y=0 .\end{array} What is the value of aa that minimizes the volume of the solid obtained by rotating the region around the xx-axis?

The volume of the solid obtained by rotating the region bounded by y=x22x y = x^2 - 2x and y=x y = x about the line y=6 y = 6, has the form abπ \frac {a}{b} \pi, where aa and bb are positive coprime integers. What is the value of a+ba+b?

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