If $f(x,y) = xy\sqrt{x^2+y^2}$ then $f(x,-y) = -f(x,y)$ and therefore \(\displaystyle \int_{-1}^1f(x,y)\,dy = 0.\) So \(\displaystyle II_{5a} = 0.\)karush said:$\textsf{a. Evaluate :}$
\begin{align*}\displaystyle
R&=[0,2] \times [-1,1]\\
II_{5a}&=\iint\limits_{R}xy\sqrt{x^2+y^2}\, dA
\end{align*}
next step?
$$\displaystyle\int_0^1 \int_{-1}^1
xy\sqrt{x^2+y^2}\, dxdy$$
Makes no difference whether it is 1 or 2. The $y$-integral is zero and therefore the double integral is zero.karush said:wait
one of the limits is [0,2]
looks like a typo in the Integral
A double integral is a type of integral used in multivariable calculus to calculate the volume under a surface defined by a function of two variables. It is represented by the symbol ∬ and is typically evaluated over a two-dimensional region in the xy-plane.
To evaluate a double integral, you first need to determine the limits of integration, which define the boundaries of the region being integrated over. Then, you can use various integration techniques, such as Fubini's Theorem or the method of polar coordinates, to solve the integral and calculate the resulting volume.
A double integral is typically denoted by ∬f(x,y)dA, where f(x,y) is the function being integrated and dA represents the infinitesimal area element in the region of integration. The limits of integration are often written as R, representing the region in the xy-plane.
Evaluating $II_{5a}$ in $R=[0,2] \times [-1,1]$ allows us to calculate the volume under a three-dimensional surface defined by the function $f(x,y)=II_{5a}$ over the two-dimensional region R. This can be useful in various applications, such as calculating the mass or center of mass of an object with varying density.
Double integrals have many practical applications, such as calculating the area of a region, finding the volume of a solid, determining the average value of a function over a region, and solving problems in physics and engineering, such as calculating work or electric flux. They are also used in probability and statistics to calculate joint probabilities and expected values.