Area Under the Curve Question - None

Question

Given, S(a) = \{(x, y) : y^2 \leq x, 0 \leq x \leq a\} and A(a) is the area of the region S(a). If \frac{A(l)}{A(4)} = \frac{2}{5} for 0 < l < 4, find the value of l.

Accepted Answer

Step 1: The region S(a) is bounded by the parabola y^2 = x and the line x = a.

Step 2: The area is given by:

A(a) = 2\int_0^a \sqrt{x}\, dx = 2\left[\frac{x^{3/2}}{3/2}\right]_0^a = \frac{4}{3}a^{3/2}

Step 3: Similarly, A(4) = \frac{4}{3}(4)^{3/2} = \frac{4}{3} \cdot 8 = \frac{32}{3}

Step 4: Using the given condition:

\frac{A(l)}{A(4)} = \frac{2}{5}

\frac{\frac{4}{3}l^{3/2}}{\frac{32}{3}} = \frac{2}{5}

\frac{l^{3/2}}{8} = \frac{2}{5}

l^{3/2} = \frac{16}{5}

\left(\frac{l}{4}\right)^{3/2} = \frac{2}{5}

\frac{l}{4} = \left(\frac{2}{5}\right)^{2/3} = \frac{4}{25}

l = 4 \cdot \frac{4}{25} = \frac{16}{25}

∴ Answer is l = \frac{16}{25}

Question:

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