Find the volume of the solid formed by rotating the region enclosed by 𝑥=0, 𝑥=1, 𝑦=0, 𝑦=7+𝑥^7 about the x-axis.

Answer 1

The first two numbers will give us the upper and lower bounds of integration. We know that, for a solid of revolution around the x-axis, the volume is π∫(a to b) (f(x))²dx. In this case, since a and b are 0 and 1 respectively and y = 7 + x⁷ = f(x), this integral becomes π∫(0 to 1) (7+x⁷)² dx = π∫(0 to 1) (49 + 14x⁷+x¹⁴)

= π[₀¹ 49x+ 14/8 x⁸ + x¹⁵/15] = π [(49*1 + 14/8 1⁸ + 1¹⁵/15) - (49*0 + 14/8*0⁸ + 0¹⁵/15)]

= π[49 +14/8 +1/15] = π[5880/120 + 210/120 + 8/120] = π[6098/120] = π[3049/60].