Question

find the area bounded by the curve y=3x-x^3 and the line y=2

Answer 1

Area enclosed:

$(3\times1)-\int_{-2}^{-\sqrt{3}}(3x-x^3)dx-\int^1_0(3x-x^3)dx+|\int^0_{-\sqrt{3}}(3x-x^3)dx|\\ \Rightarrow3-[\frac{3x^2}{2}-\frac{x^4}{4}]^{-\sqrt{3}}_{-2}-[\frac{3x^2}{2}-\frac{x^4}{4}]^{1}_{0}+|[\frac{3x^2}{2}-\frac{x^4}{4}]_{-\sqrt{3}}^{0}|\\ \Rightarrow3-[\frac{9}{2}-\frac{9}{4}-6+4]-[\frac{3}{2}-\frac{1}{4}-0+0]+|(0-0)-(\frac{9}{2}-\frac{9}{4})|\\ \Rightarrow 3-\frac{1}{4}-\frac{5}{4}+\frac{9}{4}\\ \Rightarrow \frac{15}{4} \ sq. units$