The diagram shows a curve with equation $$\(y=4 x^{\frac{1}{2}}-2 x\)$$ for $$\(x \geqslant 0\)$$, and a straight line with equation $$\(y=3-x\)$$. The curve crosses the $$\(x\)$$-axis at $$\(A(4,0)\)$$ and crosses the straight line at $$\(B\)$$ and $$\(C\)$$. Find the area of the shaded region. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................
Exam No：9709_w20_qp_11 Year：2020 Question No：12(c)
Answer:
Area of correct triangle \(=\frac{1}{2}(9-3) \times 6\)
\(\int\left(4 x^{\frac{1}{2}}-2 x\right)(\mathrm{d} x)=\left[\frac{4 x^{\frac{3}{2}}}{\frac{3}{2}}-x^{2}\right]\)
\((72-81)-\left(\frac{64}{3}-16\right)\)
\(-14 \frac{1}{3}\)
Shaded region \(=18-14 \frac{1}{3}=3 \frac{2}{3}\)
\(\int\left(4 x^{\frac{1}{2}}-2 x\right)(\mathrm{d} x)=\left[\frac{4 x^{\frac{3}{2}}}{\frac{3}{2}}-x^{2}\right]\)
\((72-81)-\left(\frac{64}{3}-16\right)\)
\(-14 \frac{1}{3}\)
Shaded region \(=18-14 \frac{1}{3}=3 \frac{2}{3}\)
Knowledge points:
1.8.3 evaluate definite integrals (Including simple cases of ‘improper’ integrals, such as)
1.8.4.1 the area of a region bounded by a curve and lines parallel to the axes, or between a curve and a line or between two curves
Solution:
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