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Q37E

Expert-verifiedFound in: Page 370

Book edition
2nd

Author(s)
James Stewart

Pages
830 pages

ISBN
9781133112280

**Find the number \(b\) such that the line \(y = b\) divides the region bounded by the curves \(y = \) \({x^2}\) and \(y = 4\) into two regions with equal area.**

The value of \(b\) is \(2.52\).

**The area under a curve between two points is found out by doing a definite integral between the two points.**

The regions \(y = {x^2},y = 4\) are divided by \(y = b\)

Integrate along the \(y\)-axis, so for \(y = {x^2}\) use \(x = \sqrt y \), which is the right half of the parabola. The right/left halves are symmetric so ignore the left half.

The red region is the integral from \(y = 0\) to \(b\), the blue region is from \(y = b\) to 4

The value of \(b\) is given by

\(\int_0^b {\sqrt y \,} dy = \int_b^4 {\sqrt y \,} dy\)

\[\left[{\frac{2}{3}{y^{\frac{3}{2}}}} \right]_0^b = \left[ {\frac{2}{3}{y^{\frac{3}{2}}}} \right]_b^4\]

\({b^{\frac{3}{2}}} - 0 = {4^{\frac{3}{2}}} - {b^{\frac{3}{2}}}\)

\(2{b^{\frac{3}{2}}} = 8\)

\({b^{\frac{3}{2}}} = 4\)

\(b = {4^{\frac{2}{3}}}\)

\( \approx 2.52\)

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