Question

Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral and using your calculator to estimate the integral. The part of the surface z = e⁻ˣ² ⁻ ʸ²that lies above the disk x²+ y²=49

Answer 1

To find the area of the surface above the given disk, we express it as a single integral in polar coordinates. We integrate the given expression and use a calculator to estimate the integral. The estimated area is approximately 3.9061 units squared.

Step-by-step explanation:

To find the area of the surface z = e⁻ˣ² ⁻ ʸ² that lies above the disk x² + y² = 49, we can express it as a single integral. We integrate the expression z = e⁻ˣ² ⁻ ʸ² over the region x² + y² ≤ 49. This gives us the area.

First, let's change the given equation from Cartesian to polar coordinates. We have
`r² = x² + y² = 49`e root of both sides, we get r = 7.

Now, we can set up the integral. The limits of integration for r are 0 and 7. The limits of integration for θ are 0 and 2π. Finally, we integrate the expression e⁻ˣ² ⁻ ʸ² with respect to r and θ.

Using a calculator to estimate the integral, we find that the area of the surface above the disk
`x² + y² = 49`tely 3.9061 units squared.