The area of the region bounded by the curve is
A) 48
B) 6
C) 24
D) 12
E) 48

Question

The area of the region bounded by the curve  is
A) 48 
B) 6 
C) 24 
D) 12 
E) 48

Quizplus · Accepted Answer

The curve given is the graph of the function $y=\sqrt{x+1}$. It starts at $(0,1)$ and ends at $(3,2)$. The area of the region bounded by this curve and the $x$-axis is given by the definite integral $\int_0^3 \sqrt{x+1} \, dx$.

We can evaluate this integral using a $u$-substitution, with $u=x+1$. Then $du=dx$ and the integral becomes $\int_1^4 \sqrt{u} \, du$. We can evaluate this using the power rule for integration: $\int_1^4 \sqrt{u} \, du = \frac{2}{3}u^{3/2} \bigg|_1^4 = \frac{2}{3}(4\sqrt{4}-1) = \frac{22}{3}$.

Therefore, the area of the region bounded by the curve is $\frac{22}{3} \approx 7.333$, which is closest to option B, 6.

The Area of the Region Bounded by the Curve