Solve \int_0^2 (7-y)/(5)-\sqrt(3-y)

Question

Evaluate the integral

\frac{46 - 30 3}{15}

Alternative Form

\approx - 0.397435

Evaluate

\int_{0}^{2} (\frac{7 - y}{5} - 3 - y) d y

Simplify the expression

\int_{0}^{2} (\frac{7}{5} - \frac{1}{5} y - 3 - y) d y

Evaluate the integral

\int (\frac{7}{5} - \frac{1}{5} y - 3 - y) d y

Rewrite the expression

\int \frac{1}{5} (7 - y - 5 3 - y) d y

Use the property of integral \int k f (x) d x = k \int f (x) d x

\frac{1}{5} \times \int (7 - y - 5 3 - y) d y

Use the property of integral \int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

\frac{1}{5} (\int 7 d y + \int - y d y + \int - 5 3 - y d y)

Calculate

\frac{1}{5} \times \int 7 d y + \frac{1}{5} \times \int - y d y + \frac{1}{5} \times \int - 5 3 - y d y

Evaluate the integral

More Steps

Evaluate

\frac{1}{5} \times \int 7 d y

Use the property of integral \int k d x = k x

\frac{1}{5} \times 7 y

Multiply the numbers

\frac{7}{5} y

\frac{7}{5} y + \frac{1}{5} \times \int - y d y + \frac{1}{5} \times \int - 5 3 - y d y

Evaluate the integral

More Steps

Evaluate

\frac{1}{5} \times \int - y d y

Use the property of integral \int k f (x) d x = k \int f (x) d x

\frac{1}{5} (- 1) \times \int y d y

Multiplying or dividing an odd number of negative terms equals a negative

- \frac{1}{5} \times \int y d y

Use the property of integral \int x^{n} d x = \frac{x ^{n + 1}}{n + 1}

- \frac{1}{5} \times \frac{y ^{1 + 1}}{1 + 1}

Simplify

More Steps

Evaluate

\frac{y ^{1 + 1}}{1 + 1}

Add the numbers

\frac{y ^{2}}{1 + 1}

Add the numbers

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

- \frac{1}{5} \times \frac{y ^{2}}{2}

Multiply the terms

- \frac{y ^{2}}{5 \times 2}

Multiply the terms

- \frac{y ^{2}}{10}

\frac{7}{5} y - \frac{y ^{2}}{10} + \frac{1}{5} \times \int - 5 3 - y d y

Evaluate the integral

More Steps

Evaluate

\frac{1}{5} \times \int - 5 3 - y d y

Use the property of integral \int k f (x) d x = k \int f (x) d x

\frac{1}{5} (- 5) \times \int 3 - y d y

Multiply the numbers

More Steps

Evaluate

\frac{1}{5} (- 5)

Multiplying or dividing an odd number of negative terms equals a negative

- \frac{1}{5} \times 5

Reduce the numbers

- 1 \times 1

Simplify

- 1

- \int 3 - y d y

Rewrite the expression

- \int (3 - y)^{\frac{1}{2}} d y

Use the substitution d y = - 1 d t to transform the integral

More Steps

Evaluate

t = 3 - y

Calculate the derivative

d t = - 1 d y

Evaluate

d y = - 1 d t

- \int (3 - y)^{\frac{1}{2}} (- 1) d t

Simplify

- \int - (3 - y)^{\frac{1}{2}} d t

Use the substitution t = 3 - y to transform the integral

- \int - t^{\frac{1}{2}} d t

Use the property of integral \int k f (x) d x = k \int f (x) d x

- (- 1) \times \int t^{\frac{1}{2}} d t

When there is - in front of an expression in parentheses change the sign of each term of the expression and remove the parentheses

1 \times \int t^{\frac{1}{2}} d t

Simplify

\int t^{\frac{1}{2}} d t

Use the property of integral \int x^{n} d x = \frac{x ^{n + 1}}{n + 1}

\frac{t ^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}

Add the numbers

More Steps

Evaluate

\frac{1}{2} + 1

Write all numerators above the least common denominator 2

\frac{1}{2} + \frac{1 \times 2}{1 \times 2}

Calculate

\frac{1}{2} + \frac{2}{2}

Add the terms

\frac{1 + 2}{2}

Add the terms

\frac{3}{2}

\frac{t ^{\frac{3}{2}}}{\frac{1}{2} + 1}

Add the numbers

More Steps

Evaluate

\frac{1}{2} + 1

Write all numerators above the least common denominator 2

\frac{1}{2} + \frac{1 \times 2}{1 \times 2}

Calculate

\frac{1}{2} + \frac{2}{2}

Add the terms

\frac{1 + 2}{2}

Add the terms

\frac{3}{2}

\frac{t ^{\frac{3}{2}}}{\frac{3}{2}}

Multiply by the reciprocal

t^{\frac{3}{2}} \times \frac{2}{3}

Use the commutative property to reorder the terms

\frac{2}{3} t^{\frac{3}{2}}

Substitute back

\frac{2}{3} (3 - y)^{\frac{3}{2}}

\frac{7}{5} y - \frac{y ^{2}}{10} + \frac{2}{3} (3 - y)^{\frac{3}{2}}

Return the limits

(\frac{7}{5} y - \frac{y ^{2}}{10} + \frac{2}{3} (3 - y)^{\frac{3}{2}})_{0}^{2}

Calculate the value

More Steps

Substitute the values into formula

\frac{7}{5} \times 2 - \frac{2 ^{2}}{10} + \frac{2}{3} (3 - 2)^{\frac{3}{2}} - (\frac{7}{5} \times 0 - \frac{0 ^{2}}{10} + \frac{2}{3} (3 - 0)^{\frac{3}{2}})

Any expression multiplied by 0 equals 0

\frac{7}{5} \times 2 - \frac{2 ^{2}}{10} + \frac{2}{3} (3 - 2)^{\frac{3}{2}} - (0 - \frac{0 ^{2}}{10} + \frac{2}{3} (3 - 0)^{\frac{3}{2}})

Removing 0 doesn't change the value,so remove it from the expression

\frac{7}{5} \times 2 - \frac{2 ^{2}}{10} + \frac{2}{3} (3 - 2)^{\frac{3}{2}} - (0 - \frac{0 ^{2}}{10} + \frac{2}{3} \times 3^{\frac{3}{2}})

Subtract the numbers

\frac{7}{5} \times 2 - \frac{2 ^{2}}{10} + \frac{2}{3} \times 1^{\frac{3}{2}} - (0 - \frac{0 ^{2}}{10} + \frac{2}{3} \times 3^{\frac{3}{2}})

Calculate

\frac{7}{5} \times 2 - \frac{2 ^{2}}{10} + \frac{2}{3} \times 1^{\frac{3}{2}} - (0 - \frac{0}{10} + \frac{2}{3} \times 3^{\frac{3}{2}})

1 raised to any power equals to 1

\frac{7}{5} \times 2 - \frac{2 ^{2}}{10} + \frac{2}{3} \times 1 - (0 - \frac{0}{10} + \frac{2}{3} \times 3^{\frac{3}{2}})

Divide the terms

\frac{7}{5} \times 2 - \frac{2 ^{2}}{10} + \frac{2}{3} \times 1 - (0 - 0 + \frac{2}{3} \times 3^{\frac{3}{2}})

Reduce the numbers

\frac{7}{5} \times 2 - \frac{2 ^{2}}{10} + \frac{2}{3} \times 1 - (0 - 0 + 2 \times 3^{\frac{1}{2}})

Divide the terms

More Steps

Evaluate

\frac{2 ^{2}}{10}

Rewrite the expression

\frac{2 ^{2}}{2 \times 5}

Reduce the fraction

\frac{2}{5}

\frac{7}{5} \times 2 - \frac{2}{5} + \frac{2}{3} \times 1 - (0 - 0 + 2 \times 3^{\frac{1}{2}})

Multiply the numbers

More Steps

Evaluate

\frac{7}{5} \times 2

Multiply the numbers

\frac{7 \times 2}{5}

Multiply the numbers

\frac{14}{5}

\frac{14}{5} - \frac{2}{5} + \frac{2}{3} \times 1 - (0 - 0 + 2 \times 3^{\frac{1}{2}})

Any expression multiplied by 1 remains the same

\frac{14}{5} - \frac{2}{5} + \frac{2}{3} - (0 - 0 + 2 \times 3^{\frac{1}{2}})

Removing 0 doesn't change the value,so remove it from the expression

\frac{14}{5} - \frac{2}{5} + \frac{2}{3} - 2 \times 3^{\frac{1}{2}}

Calculate the sum or difference

More Steps

Evaluate

\frac{14}{5} - \frac{2}{5} + \frac{2}{3}

Subtract the numbers

\frac{12}{5} + \frac{2}{3}

Write all numerators above the least common denominator 15

\frac{12 \times 3}{5 \times 3} + \frac{2 \times 5}{3 \times 5}

Calculate

\frac{36}{15} + \frac{10}{15}

Add the terms

\frac{36 + 10}{15}

Add the terms

\frac{46}{15}

\frac{46}{15} - 2 \times 3^{\frac{1}{2}}

\frac{46}{15} - 2 \times 3^{\frac{1}{2}}

Use a^{\frac{m}{n}} = n a^{m} to transform the expression

\frac{46}{15} - 23

Solution

\frac{46 - 30 3}{15}

Alternative Form

\approx - 0.397435

Show Solution