Question

Find the volume of the solid formed by rotating the region bounded by the given curves about the indicated axis. y = x3/4, x = 0, y = 1 (in the first quadrant); about the y-axis

Answer 1

Answer:

7.92 cubic units

Step-by-step explanation:

The volume formed is V = ∫πx²dy

Now, since y = x³/4, x = ∛(4y). Also if x = 0, y = 0³/4 = 0 and the curve intersects the line y = 1. So the limits of integration are y = 0 to y = 1

So, V = ∫₀¹πx²dy

= ∫₀¹π[ ∛(4y)]²dy

= π(∛4)²∫₀¹[ ∛(y]²dy

= π(∛4)²∫₀¹y^³/₂dy

= π(∛4)²[y^⁵/₂]₀¹

= π(∛4)²[1^⁵/₂ - 0^⁵/₂]

= π(∛4)²[1 - 0]

= π(∛4)²

= 7.92 cubic units