Solve ∫ (from 0.5 to 1) of (9800*0.5*(4-y)) wrt y

$$\int _ { 0.5 } ^ { 1 } ( 9800 \cdot 0.5 \cdot ( 4 - y ) ) d y$$

Evaluate

$$7962.5$$

Solution Steps

Multiply 9800 and 0.5 to get 4900.

$$\int _{0.5}^{1}4900\left(4-y\right)\mathrm{d}y$$

Use the distributive property to multiply 4900 by 4-y.

$$\int _{0.5}^{1}19600-4900y\mathrm{d}y$$

Evaluate the indefinite integral first.

$$\int 19600-4900y\mathrm{d}y$$

Integrate the sum term by term.

$$\int 19600\mathrm{d}y+\int -4900y\mathrm{d}y$$

Factor out the constant in each of the terms.

$$\int 19600\mathrm{d}y-4900\int y\mathrm{d}y$$

Find the integral of 19600 using the table of common integrals rule \int a\mathrm{d}y=ay.

$$19600y-4900\int y\mathrm{d}y$$

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 -4900 times \frac{y^{2}}{2}.

$$19600y-2450y^{2}$$

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.

$$19600\times 1-2450\times 1^{2}-\left(19600\times 0.5-2450\times 0.5^{2}\right)$$

Simplify.

$$7962.5$$