Final answer:
To calculate the double integral ∬ R xsec²(y) dA, we integrate with respect to y first, and then integrate the resulting expression with respect to x.
Step-by-step explanation:
The student has asked to calculate the double integral ∫∫ R x sec²(y) dA, where R is the rectangular region with 0 ≤ x ≤ 8 and 0 ≤ y ≤ 4. To proceed with the integral, we integrate x with respect to x first and then integrate sec²(y) with respect to y.
The integration over x from 0 to 8 is simple, as it gives us a factor of x²/2 from 0 to 8, which simplifies to 32. The integration of sec²(y) from 0 to 4 is the integral of the derivative of tan(y), which evaluates from tan(0) to tan(4).
The calculation proceeds as:
- Integrate x to get (x²/2) from 0 to 8 giving us 32.
- Integrate sec²(y) to get tan(y) from 0 to 4.
- Calculate the product of these two results which gives us 32 * (tan(4) - tan(0)).
- Since tan(0) is 0 and we only need the value of tan(4), we multiply 32 by tan(4).
Since the value of tan(4) is not readily available and exceeds the scope of typical high school trigonometry values, it's understood that this value would be obtained using a calculator or a table of trigonometric values. However, it is not necessary to find this empirical value to complete the double integral since it's clear that the product of 32 and tan(4), even without its exact value, is not an integer and hence does not match any of the provided multiple-choice options (A) 64, (B) 96, (C) 128, (D) 192.