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

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.

Finding the intersection points

Identifying the areas and approach

Integrating from the intersection points

Graphing the functions

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

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.

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

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

Using numerical methods like Simpson's Rule over assuming analytical ones may fail to account for precise symmetry and curve interactions.

Applying a single integral from -a to a without splitting it, since the intersection points mean the same area is not covered more than once.

Setting up a symmetrical integration from 0 to a since both functions are even, and then doubling the result to capture the entire area on the interval.

Assuming that the functions intersect at origin only and thus computing from -a neglectful of additional intersections.