Solve ∫ (from 0 to 1) of intx^2+y^2-5

$$\int _ { 0 } ^ { 1 } \int x ^ { 2 } + y ^ { 2 } - 5$$

Evaluate

$$\frac{y^{2}}{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \int x^{2}+y^{2}-5\mathrm{d}x\mathrm{d}x$$

Simplify.

$$\frac{x^{4}}{12}+\frac{y^{2}x^{2}}{2}-\frac{5x^{2}}{2}+Сx$$

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.

$$\frac{1^{4}}{12}+\frac{1}{2}y^{2}\times 1^{2}-\frac{5}{2}\times 1^{2}+С\times 1-\left(\frac{0^{4}}{12}+\frac{1}{2}y^{2}\times 0^{2}-\frac{5}{2}\times 0^{2}+С\times 0\right)$$

Simplify.

$$\frac{y^{2}}{2}$$

Differentiate w.r.t. y

$$y$$