Applications of Integration

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

Approximation assumes continuity but not necessarily differentiability across all segments after intersections

Summation approximation requires only positive values for either function over each subinterval defined by intersections

Integral bounds must adjust dynamically based solely on maximum values between successive intersections

Each individual shape's area should not overlap nor leave gaps between them across intervals defined by consecutive intersections

The x-coordinate of the point is the same in both curves

The x-coordinate and y-coordinate of the point are the same in both curves

The y-coordinate of the point is the same in both curves

The x-coordinate and y-coordinate of the point are different in both curves

The uppermost function relative to the x-axis.

The rightmost function related to y-axis.

The linear function before any quadratic functions.

The lowermost function relative to the x-axis.

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

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

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

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

Implementing trapezoidal rule across entire domain without considering curve behavior changes or intersection nuances between endpoints.

Failing to use separative definite integrals based on point-to-point analysis leading up to $(0,1)$ due to identical start/end points masking internal intersections.

Using a single application of integration by parts for entire region assuming $f(x)$ is always above $g(x)$.

Assuming that since functions have identical endpoints there are no internal intersections thus requiring only one integral calculation.

$\sin{x}$ and $\cos{2x}$ never cross on a specified interval.

Integration can only be done graphically in this scenario.

The intersections may not yield exact algebraic solutions.

The antiderivative of $\sin{x} - \cos{2x}$ cannot be found using basic calculus techniques.

Finding the intersection points

Identifying the areas and approach

Integrating from the intersection points

Graphing the functions

Limits of integration based on points of intersection

Slope-intercept form of one curve only

Axis of symmetry for parabolic regions

Derivative of a product of both functions