Solve ∫ (from 0 to 1) of sqrt{1-(sqrt{2}y^1/2)^2}

$$\int _ { 0 } ^ { 1 } \sqrt { 1 - ( \sqrt { 2 } y ^ { 1 / 2 } ) ^ { 2 } }$$

Evaluate

$$\sqrt{1-2y}$$

Solution Steps

Expand \left(\sqrt{2}y^{\frac{1}{2}}\right)^{2}.

$$\int _{0}^{1}\sqrt{1-\left(\sqrt{2}\right)^{2}\left(y^{\frac{1}{2}}\right)^{2}}\mathrm{d}x$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{2} and 2 to get 1.

$$\int _{0}^{1}\sqrt{1-\left(\sqrt{2}\right)^{2}y^{1}}\mathrm{d}x$$

The square of \sqrt{2} is 2.

$$\int _{0}^{1}\sqrt{1-2y^{1}}\mathrm{d}x$$

Calculate y to the power of 1 and get y.

$$\int _{0}^{1}\sqrt{1-2y}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int \sqrt{1-2y}\mathrm{d}x$$

Find the integral of \sqrt{1-2y} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\sqrt{1-2y}x$$

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.

$$\left(1-2y\right)^{\frac{1}{2}}\times 1-\left(1-2y\right)^{\frac{1}{2}}\times 0$$

Simplify.

$$\sqrt{1-2y}$$

Differentiate w.r.t. y

$$-\frac{1}{\sqrt{1-2y}}$$