 ### Find all the second partial derivatives of v=xy/x-y.

This question aims to find all the second-order partial derivatives of the given function. The derivative of a function with more than one variable with respect to one of the variables present in the function while treating the other variables as constant is called a partial derivative of that function. In other words, when the function input is […]

### Find the average value of f over the given rectangle. f(x,y)= x^2y. R has vertices (-1,0),(-1,5),(1,5),(1,0)

The objective of this question is to find the average value of the function over the given region which is a rectangle. The average value of a bounded set of numbers is described as the total of the numbers divided by the number of numbers. In other words, a function’s average value is the average height […]

### As viewed from above, a swimming pool has the shape of the ellipse x^2/2500+y^2/900=1, where x and y are measured in feet. The cross sections perpendicular to the x-axis are squares. Find the total volume of the pool.

This question aims to find the volume of the pool having square cross-sections perpendicular to the $x-$axis. The numerical measure of a three-dimensional solid is said to be the volume. The concept is similar to the measurement of the area in a two-dimensional region. Usually, some of the fundamental geometric concepts are used to calculate […]

### Use a double integral to find the area of the region. The region inside the cardioid r = 1 + cos(θ) and outside the circle r = 3 cos(θ).

This question aims to find the area of the region described by the given equations in polar form. A two-dimensional plane with a curve whose shape is like a heart is said to be a cardioid. This term is derived from a Greek word that means “heart.” Therefore, it is known as a heart-shaped curve. […]

### For which positive integers k is the following series convergent?

(sumlimits_{n=1}^{infty}dfrac{(n!)^2}{(kn)!}) This question aims to find the value of the positive integer $k$, for which the given series is convergent. A series in mathematics is a representation of the procedure of adding infinite quantities sequentially to a given starting quantity. The series analysis is an important part of calculus and its generalization such as mathematical analysis. A convergent series […]