Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. $\int _ { 0 } ^ { 9 } \sqrt { 81 - t ^ { 2 } } d t$
A) $\frac { 81 } { 4 } \pi$
B) $\frac { 81 } { 8 } \pi$
C) $\frac { 81 } { 2 } \pi$
D) $\frac { 81 } { 4 }$
E) $81 \pi$

Question

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. $\int _ { 0 } ^ { 9 } \sqrt { 81 - t ^ { 2 } } d t$ 
A) $\frac { 81 } { 4 } \pi$
B) $\frac { 81 } { 8 } \pi$
C) $\frac { 81 } { 2 } \pi$
D) $\frac { 81 } { 4 }$
E) $81 \pi$

Quizplus · Accepted Answer

The integral $\int _ { 0 } ^ { 9 } \sqrt { 81 - t ^ { 2 } } d t$ represents a quarter circle with radius 9 (since $81 = 9^2$), so the area is $\frac{1}{4} \pi r^2 = \frac{1}{4} \pi (9)^2 = \frac{81}{4} \pi$.

Sketch the Region Whose Area Is Given by the Definite ∫0981−t2dt\int _ { 0 } ^ { 9 } \sqrt { 81 - t ^ { 2 } } d t∫09​81−t2​dt

Sketch the Region Whose Area Is Given by the Definite $\int _ { 0 } ^ { 9 } \sqrt { 81 - t ^ { 2 } } d t$