Set up the double integral required to find the moment of inertia I, about the given line, of the lamina bounded by the graphs of the equations. Use a computer algebra system to evaluate the double integral. x + y = b2 p = k line: x = a (a > b) b. I = 2 knb (2 + 4a ) dy dx =

Q: Evaluate the line integral, where C is the given curve. xy dx + (x − y) dy, where C consists of…

A: In the above line integral there are two segments which needto be seperated. First (0,0) to (1,0) so…

Q: Compute the line integral of the scalar function f(x, y) = /1+9xy over the curve y = x³ for 0 < x <…

A:

Q: Calculate the double integral. x sec?{v) dA, R- {x, v) I0 sx s 6, 0 s ys} x sec?(y) dA, =

A:

Q: evaluate the line integral along the curve C. Scv+z) d s, C is the straight-line segment x = 0, y =…

A: To Evaluate∫C(y+z)dswhere C is straight line segmentx=0  ----(i)y=3-t -----(ii)z=t…

Q: Express the triple integral in cylindrical coordinates. x² + y2 f(x, y, z) dz dy dx 3x – x2 f(r…

A:

Q: Evaluate the integral by changing to cylindrical coordinates. 16 - y2 xz dz dx dy 16-y2 Vx2+

A: The given integral is                      ∫-44∫-16-y216-y2∫x2+y26xz dz dx dy

Q: Find the value of integral ((x +y +:)ds where C is a part of helix parameterized by r(t) = (cost,…

A: ds = [ (dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2 ]^(1/2) dt

Q: Evaluate the integral by changing to cylindrical coordinates. V 16 – x2 r 16 – x2 - y2 x² + y2 dz dy…

A:

Q: Calculate the double integral || (x + y)?ex -ý dx dy where R is the square with vertices (2, 0), (0,…

A:

Q: Calculate the double integral. sec?ty) da, {(x, v) I 0 s x s 2, 0 s y s - {w, )losxs2,0sys

A:

Q: Calculate the integral (x – y) dx + (x + y) dy I = x2 + y2 where L is along the curve y = cos x,…

A:

Q: Evaluate the integral by changing to cylindrical coordinates.

A: Given triple integral as,

Q: Find the potential function of the field F (x, y, z) = i+j+k. Assume that the constant of…

A:

Q: compute the line integral f(x, y) ds for the given function and path or curve. f(x, y) = x – y, the…

A: Given: fx,y=x-y To Find: Line integral ∫Cfx,yds over the unit semi-circle x2+y2=1, y≥0

Q: Use cylindrical coordinates to evaluate the integral below. F √x² + y²dv, where E is the region that…

A: We can solve this as follows:

Q: f(x, y, z) dV = JE

A:

Q: Evaluate the integral by changing to cylindrical coordina xz dz dx dy -2J-/4-y² x+y²

A: Introduction:   Let f(x,y,z) be a three-variable continuous function defined over a solid Q. The…

Q: Consider the following line integral along the given positively oriented curve. | yeš dx + 2ex dy,…

A:

Q: Evaluate the integral f. x'y°dy – x³y'dx, where C is the square determined by the lines x = 0, y =…

A: We have to evaluate the given integral.

Q: Find the line integral of ƒ(x, y) = x - y + 3 along the curver(t) = (cos t)i + (sin t)j, 0 &lt;_ t…

A:

Q: Compute the line integral of the scalar function f(x, y) = /1 + 9xy over the curve y = x' for 0 < x…

A:

Q: Calculate the integral of f(x,y,z)=3x2 +3y2 +z4 over the curve c(t)=(cost,sint,t) for 0≤t≤π. ∫c…

A:

Q: Compute the line integral of the scalar function f(x, y) = over the curve y = x° for 0 <x <1 /1+ 9xy…

A:

Q: Use a calculator or CAS to evaluate the line integral correct to four decimal places. F. dr, where…

A:

Q: Evaluate the integral by changing to cylindrical coordinates. xz dz dx dy V4- ya Vx² + y2

A: We will solve the problem.

Q: Compute the line integral of the scalar function f(x, y) = /1+ 9ry over the curve y = x° for 0 <¤ <…

A:

Q: Verify Stokes's Theorem by evaluating F(x, y, z) = xyzi + yj + zk S: 8x + 8y + z = 16, first octant…

A:

Q: Find the value of integral (x+y+:) ds where C is a part of helix parameterized by r(t) = (cost, sin…

A:

Q: Evaluate the surface integral /F- ds for the veetor field F = (y. -r, 2:), where S is the hemisphere…

A: Surface integration

Q: Determine the line integral of a scalar function z = x cos y + In(xy) where x(t) = t? and y(t) = 3…

A: To determine the line integral of the below scalar function where xt=t2 and yt=3+t bounded by the…

Q: Evaluate the line integral using Green's Theorem and check the answer by evaluating it directly.…

A: Here we have to use green's theorem.

Q: 9. Evaluate the line integral of F = (2xy + 1,x² + 2yz, y? – 2) over the region C parametrized by…

A:

Q: Evaluate the integral by changing to cylindrical coordinates. 100 y2 13 10 xz dz dx dy Vx2 + y2 100…

A:

Q: .3 T || f(r, e) r de dr Jo 0.

A: Solved below

Q: Evalute The Line integral of The Fiel f=xi+zj+yK o ver The helix rz cost i +sintjt tK for osts I

A:

Q: Evaluate the iterated integral by changing to cylindrical coordinates. V16 - x (x²+ y?)3/2 dy dx dz…

A:

Q: Evaluate the line integral, where C is the given curve. z2 dx + x2 dy + y? dz, C is the line segment…

A:

Q: Evaluate the line integral, where C is the given curve; rds; where C:x =t°,y = t, 0<t<1 (ii) ryds…

A: The solution is given as

Q: Calculate the line integral ∫Cyxdx+dy over the curve y=lnx in the interval 1≤x≤e (Figure 4).

A:

Q: Find the line integral of ƒ(x, y) = x - y + 3 along the curve r(t) = (cos t)i + (sin t)j, 0&lt;=…

A:

Q: Evaluate the line integral along the curve C. ds, C is the curve r(t) = (-1 - t) i - j+(-1 - t) k,…

A:

Q: Evaluate the integral by changing to cylindrical coordinates. 16-1 c16-r² -y? Vx2 + y? dz dy dx =…

A: Given: The given integral is, ∫-44∫016-x2∫016-x2-y2x2+y2dzdydx. The objective is to evaluate the…

Q: Evaluate the integral by changing to cylindrical coordinates. 16 - y2 xz dz dx dy 16 - y2 x² + yZ

A: Here, the given limits are z=x2+y2   to   z=7. Put x2+y2=r2 implies, z=r The limits of z in…

Q: Evaluate the line integral using Green's Theorem and check the answer by evaluating it directly. $…

A:

Q: Evaluate the line integral, where C is the given curve. z? dx + x2 dy + y² dz, C is the line segment…

A:

Q: valuate the line integral J xev% ds where C is the line segment from > (1, 1, 1). This yields:

A:

Q: Evaluate the double integral: L cos(") VF dy da. Hint: Change the order of integration to dxdy.

A: order of change of integrals

Q: V9 - y2 xz dz dx dy -V 9 – y2 'V x² + y2

A:

Q: Calculate the double integral. sec) da, R={cx, v) 1 0 s*s2,0s ys} sec2(y) dA, R = = {(x, y) | 0 < x…

A:

Q: Express the integral as an equivalent integral in which the z-integra is performed first, the…

A: Given integral ∫036∫06-x∫0zfx,y,z dydzdx. Here the integration is in the order - y integration is…

Compound Probability

Compound probability can be defined as the probability of the two events which are independent. It can be defined as the multiplication of the probability of two events that are not dependent. 

Tree diagram

Probability theory is a branch of mathematics that deals with the subject of probability. Although there are many different concepts of probability, probability theory expresses the definition mathematically through a series of axioms. Usually, these axioms express probability in terms of a probability space, which assigns a measure with values ranging from 0 to 1 to a set of outcomes known as the sample space. An event is a subset of these outcomes that is described. 

Conditional Probability

By definition, the term probability is expressed as a part of mathematics where the chance of an event that may either occur or not is evaluated and expressed in numerical terms. The range of the value within which probability can be expressed is between 0 and 1. The higher the chance of an event occurring, the closer is its value to be 1. If the probability of an event is 1, it means that the event will happen under all considered circumstances. Similarly, if the probability is exactly 0, then no matter the situation, the event will never occur.