Question

Compute the volume of an object if its base is a circle with radius 6 and cross sections perpendicular to the base are squares. Enter the exact value of the answer.

Answer 1

Answer:

$V=1152$

Step-by-step explanation:

Given that,

   The base of the object is a circle whose radius is 6.

     i.e   $x^2+y^2=36$

         $\Rightarrow y^2=36-x^2\\\Rightarrow y=\sqrt{36-x^2}$

Now, Volume $=\int\limits^6_{-6} ({2\sqrt{36-x^2})^2} \, dx$        $[\because \text{one side of square}=2\sqrt{36-x^2} \\ \quad \quad A=S^2=(2\sqrt{36-x^2})^2 ]$

               $\Rightarrow V=\int\limits^6_{-6} {4(36-x^2)} \, dx$

               $\Rightarrow V=\int\limits^6_{-6} {(144-4x^2)} \, dx$

                $\Rightarrow V=[144x-\frac{4x^3}{3}]^6_{-6} \, dx$

                         $=(864-288)-(-864+288)$

                         $=576+576$

                         $=1152$

                 $\Rightarrow V=1152$