Question

make a sketch to show a case in which the area bounded by two curves is most easily found by integrating with respect to x.
which of the following is a case in which the area bounded by two curves is most easily found by integrating with respect to x?
oa
ob
oc
od

Explanation:

Step1: Analyze the concept of area between curves

The formula for the area between two curves \(y = f(x)\) and \(y = g(x)\) from \(x=a\) to \(x = b\) is \(A=\int_{a}^{b}\vert f(x)-g(x)\vert dx\). For this formula to be easily applicable, the curves should be such that one is clearly above the other for the entire interval of integration with respect to \(x\).

Step2: Examine Option A

In Option A, the curves are set - up in a way that for the relevant interval (the interval that bounds the region), one curve is above the other. So, we can directly apply the formula \(A=\int_{a}^{b}(y_{upper}-y_{lower})dx\) without having to split the integral into multiple parts based on \(x\).

Step3: Examine Option B

In Option B, if we try to integrate with respect to \(x\), we would have to split the integral into two parts. Because the "upper" and "lower" curves swap their positions within the interval that bounds the region.

Step4: Examine Option C

In Option C, if we consider integrating with respect to \(x\), the curves are not well - behaved for a simple \(dx\) integration. The relationship between the curves (which one is on top) is not straightforward over a single interval of \(x\). It might be more complicated to set up the integral \(\int_{a}^{b}(y_{upper}-y_{lower})dx\) compared to Option A.

Step5: Examine Option D

In Option D, similar to Option B and C, if we try to integrate with respect to \(x\) to find the area between the curves, we may need to split the integral into multiple sub - integrals based on the \(x\) - values where the relative positions (upper and lower) of the curves change.

Answer:

A.