Applications of Integration

Determine all points of intersection between the functions.

Find all asymptotes for both functions involved.

Calculate derivatives of both functions.

Compute antiderivatives for each function separately.

Add up the definite integrals of both functions from their respective intersections with the y-axis.

Integrate $(f(x) - g(x))$ only from the first to second intersection point.

Integrate $(g(x) - f(x))$ from the smallest to largest intersection points, considering separate integrals if necessary.

Find one point of intersection and double integrate $(f(x))$ from that point.

Only find the intersection points

Graph the functions, identify the areas and approach, find the intersection points, integrate from the intersection points

Only identify the areas and approach

Only graph the functions

Employing substitution methods to simplify the domain for integration along with interval bounds identification.

Relying solely on graphical methods for approximation without any formal calculus-based analysis.

Utilizing the differential of two functions to locate their intersection points before integration.

Performing numerical integration techniques, followed by solving a pair or set of equations for the precise intersection points.

2 + \sqrt{3}

$\frac{3}{2} - \sqrt{2}$

$\frac{3}{2} + \sqrt{2}$

$\frac{\pi}{4} + 1$

A cubic polynomial with two inflection points.

A trigonometric function that crosses the x-axis three times.

A quadratic polynomial with one positive root.

A quartic polynomial with two real positive roots and one negative root.

One

Two

It depends on the specific curves

Three or more

Rolle's Theorem

The Intermediate Value Theorem

The Mean Value Theorem

The Fundamental Theorem of Calculus

Integrate from the intersection points

Find the intersection points

All of the above

Identify the areas and approach

Integrate (right function - left function) with respect to x from the y-values of the intersection points

Integrate (right function - left function) with respect to y from the y-values of the intersection points

Integrate (top function - bottom function) with respect to y from the x-values of the intersection points

Integrate (top function - bottom function) with respect to x