Assessment: Applications of Integration

Evaluate ∫₀² 4x³ dx.

Find the area under y = 2x, from x = 0 to x = 3.

Find the area between f(x) = 4 and g(x) = x², between their intersection points.

Find the volume when the region under y=3, from x=0 to x=2, is rotated around the x-axis.

Find the volume when the region between R(x)=√x (outer) and r(x)=x/2 (inner), from x=0 to x=4, is rotated around the x-axis.

Evaluate ∫₋₁¹ (3x² + 1) dx.

A factory's production rate is r(t) = 50 + 2t units per day. What does ∫₀¹⁰ r(t) dt represent?

A ceramicist designs a vase by rotating a profile curve y=f(x) around the x-axis (the vase's central axis). Which formula computes the vase's volume?

Find the area enclosed between f(x) = 8−x² and g(x) = x².

Find the volume when the region between R(x)=x+1 (outer) and r(x)=x (inner), from x=0 to x=2, is rotated around the x-axis.