Solve ∫ (from 0 to 4) of 8-4sqrt[3]{y}-2y-sqrt[4]{y wrt y

$$\int_{ 0 }^{ 4 } 8-4 \sqrt[3]{ y } -2y- \sqrt[ 4 ]{ y } d y$$

Evaluate

$$-\frac{16\sqrt{2}}{5}+16-12\times 2^{\frac{2}{3}}\approx -7.574296023$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 8-4\sqrt[3]{y}-2y-\sqrt[4]{y}\mathrm{d}y$$

Integrate the sum term by term.

$$\int 8\mathrm{d}y+\int -4\sqrt[3]{y}\mathrm{d}y+\int -2y\mathrm{d}y+\int -\sqrt[4]{y}\mathrm{d}y$$

Factor out the constant in each of the terms.

$$\int 8\mathrm{d}y-4\int \sqrt[3]{y}\mathrm{d}y-2\int y\mathrm{d}y-\int \sqrt[4]{y}\mathrm{d}y$$

Find the integral of 8 using the table of common integrals rule \int a\mathrm{d}y=ay.

$$8y-4\int \sqrt[3]{y}\mathrm{d}y-2\int y\mathrm{d}y-\int \sqrt[4]{y}\mathrm{d}y$$

Rewrite \sqrt[3]{y} as y^{\frac{1}{3}}. Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y^{\frac{1}{3}}\mathrm{d}y with \frac{y^{\frac{4}{3}}}{\frac{4}{3}}. Simplify. Multiply -4 times \frac{3y^{\frac{4}{3}}}{4}.

$$8y-3y^{\frac{4}{3}}-2\int y\mathrm{d}y-\int \sqrt[4]{y}\mathrm{d}y$$

Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y\mathrm{d}y with \frac{y^{2}}{2}. Multiply -2 times \frac{y^{2}}{2}.

$$8y-3y^{\frac{4}{3}}-y^{2}-\int \sqrt[4]{y}\mathrm{d}y$$

Rewrite \sqrt[4]{y} as y^{\frac{1}{4}}. Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y^{\frac{1}{4}}\mathrm{d}y with \frac{y^{\frac{5}{4}}}{\frac{5}{4}}. Simplify. Multiply -1 times \frac{4y^{\frac{5}{4}}}{5}.

$$8y-3y^{\frac{4}{3}}-y^{2}-\frac{4y^{\frac{5}{4}}}{5}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$8\times 4-3\times 4^{\frac{4}{3}}-4^{2}-\frac{4}{5}\times 4^{\frac{5}{4}}-\left(8\times 0-3\times 0^{\frac{4}{3}}-0^{2}-\frac{4}{5}\times 0^{\frac{5}{4}}\right)$$

Simplify.

$$16-12\times 2^{\frac{2}{3}}-\frac{16\sqrt{2}}{5}$$