Question

The area (in sq. units) of the region enclosed by \(\{(x,y)|y\leq x^2,xy\leq 8,y\geq 1\}\) is...

• A. \(8log2-\frac{14}{3}\)
• B. \(8log2+\frac{7}{3}\)
• C. \(16log2+\frac{7}{3}\)
• D. \(16log2-\frac{14}{3}\)

Hint:

Find the intersection points and integrate between the curves in the first quadrant.

Answer 1

The Correct Option is D

Step 1: Use Horizontal Strips:
Take the first-quadrant region. At height $y$, the condition $y\leq x^2$ gives $x\geq\sqrt y$, and $xy\leq8$ gives $x\leq\dfrac8y$. The strip has width $\dfrac8y-\sqrt y$.

Step 2: Find the Range of y:
The strip exists only when $\sqrt y\leq\dfrac8y$, i.e. $y^{3/2}\leq8$, i.e. $y\leq4$. With $y\geq1$, the heights run from 1 to 4.

Step 3: Integrate:
\[ A=\int_1^4\left(\frac8y-\sqrt y\right)dy=8\ln4-\frac23\left(4^{3/2}-1\right)=16\ln2-\frac{14}3 \]
This agrees with option (D).

Final Answer:
Option (D). \[ \boxed{\text{(D) } 16\log2-\frac{14}{3}} \]