Enter a problem...
Calculus Examples
,
Step 1
Step 1.1
Eliminate the equal sides of each equation and combine.
Step 1.2
Solve for .
Step 1.2.1
Move all terms containing to the left side of the equation.
Step 1.2.1.1
Add to both sides of the equation.
Step 1.2.1.2
Add and .
Step 1.2.2
Factor out of .
Step 1.2.2.1
Factor out of .
Step 1.2.2.2
Factor out of .
Step 1.2.2.3
Factor out of .
Step 1.2.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 1.2.4
Set equal to .
Step 1.2.5
Set equal to and solve for .
Step 1.2.5.1
Set equal to .
Step 1.2.5.2
Add to both sides of the equation.
Step 1.2.6
The final solution is all the values that make true.
Step 1.3
Evaluate when .
Step 1.3.1
Substitute for .
Step 1.3.2
Multiply by .
Step 1.4
Evaluate when .
Step 1.4.1
Substitute for .
Step 1.4.2
Multiply by .
Step 1.5
The solution to the system is the complete set of ordered pairs that are valid solutions.
Step 2
Reorder and .
Step 3
The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.
Step 4
Step 4.1
Combine the integrals into a single integral.
Step 4.2
Multiply .
Step 4.2.1
Multiply by .
Step 4.2.2
Multiply by .
Step 4.3
Add and .
Step 4.4
Split the single integral into multiple integrals.
Step 4.5
Since is constant with respect to , move out of the integral.
Step 4.6
By the Power Rule, the integral of with respect to is .
Step 4.7
Combine and .
Step 4.8
Since is constant with respect to , move out of the integral.
Step 4.9
By the Power Rule, the integral of with respect to is .
Step 4.10
Simplify the answer.
Step 4.10.1
Combine and .
Step 4.10.2
Substitute and simplify.
Step 4.10.2.1
Evaluate at and at .
Step 4.10.2.2
Evaluate at and at .
Step 4.10.2.3
Simplify.
Step 4.10.2.3.1
Raise to the power of .
Step 4.10.2.3.2
Raising to any positive power yields .
Step 4.10.2.3.3
Cancel the common factor of and .
Step 4.10.2.3.3.1
Factor out of .
Step 4.10.2.3.3.2
Cancel the common factors.
Step 4.10.2.3.3.2.1
Factor out of .
Step 4.10.2.3.3.2.2
Cancel the common factor.
Step 4.10.2.3.3.2.3
Rewrite the expression.
Step 4.10.2.3.3.2.4
Divide by .
Step 4.10.2.3.4
Multiply by .
Step 4.10.2.3.5
Add and .
Step 4.10.2.3.6
Raise to the power of .
Step 4.10.2.3.7
Raising to any positive power yields .
Step 4.10.2.3.8
Cancel the common factor of and .
Step 4.10.2.3.8.1
Factor out of .
Step 4.10.2.3.8.2
Cancel the common factors.
Step 4.10.2.3.8.2.1
Factor out of .
Step 4.10.2.3.8.2.2
Cancel the common factor.
Step 4.10.2.3.8.2.3
Rewrite the expression.
Step 4.10.2.3.8.2.4
Divide by .
Step 4.10.2.3.9
Multiply by .
Step 4.10.2.3.10
Add and .
Step 4.10.2.3.11
Combine and .
Step 4.10.2.3.12
Multiply by .
Step 4.10.2.3.13
To write as a fraction with a common denominator, multiply by .
Step 4.10.2.3.14
To write as a fraction with a common denominator, multiply by .
Step 4.10.2.3.15
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 4.10.2.3.15.1
Multiply by .
Step 4.10.2.3.15.2
Multiply by .
Step 4.10.2.3.15.3
Multiply by .
Step 4.10.2.3.15.4
Multiply by .
Step 4.10.2.3.16
Combine the numerators over the common denominator.
Step 4.10.2.3.17
Simplify the numerator.
Step 4.10.2.3.17.1
Multiply by .
Step 4.10.2.3.17.2
Multiply by .
Step 4.10.2.3.17.3
Add and .
Step 5