Find the area enclosed by the loop in the graph of the curve E: x(t) = t?, y(t) = t³ – 3t by evaluating an integral of the P -t1 form: | - y(t) · x'(t) dt for a to suitable choices of to and t (where t, < t, and Pto = Pt, = P). %3| B: x ( t) = ? , y ( t) = t³ – 31

Q: Which of the following integrals represents the length of the curve y = cos(x), 0 < x < 2n? A. V1 +…

A: We have to find integral that represents the length of given curve.

Q: Calculate the integral O (y – e*) dx + (3x + e') dy where L is along the curve x + y´ : = /x2 + y² -…

A:

Q: Calculate the integral of f(æ, y) = 10x over the region D bounded above by y = x(2 – x) and below by…

A:

Q: If | f(z)dx f(x)dx + what are the bounds of integration for the first integral? a = and b =

A: Given that ∫abfxdx=∫−47fxdx+∫78fxdx−∫−41fxdx

Q: Evaluate the double integral Is- (2 - 2 2) dxdy where S is the region in the 4th quadrant (y <0, z2…

A: Given the double integral and equation of curve I=∬Sx2−2x3dxdy And y−3x,y=−2x2

Q: The improper integral A e B 0 C D) 1 + ∞ [** 0 e-x dx is

A: The given improper integral is: I=∫0+∞e-xdx

Q: Question included in image

A: The region of Integration can be defined as

Q: Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's rule to approximate the integral In(x) dx…

A: We use trapezoidal rule ,midpoint rule,simpson's rule to approximate the given integral

Q: (a) The Composite Trapezoidal Rule applied to the integral - [ f(z)dx I = gives the error b - a, 4h³…

A:

Q: Evaluate the integral ,(1 – x – 4x3 + 2x5) dx a) Analytically b) Single application of trapezoidal…

A:

Q: If the order of integration is reversed for the double integral J e* dxdy, the new et? limits will…

A:

Q: Evaluate the double integral: To L 0 y x7 +1 y¹/3 x7 dx dy.

A: The given integral is: I=∫08∫y132yx7+1dxdy   So, The limit of the x varies from y13 to 2. The limit…

Q: Sketch the region of integration corresponding to y²e= drdy, (log æ)3 J = and evaluate J.

A: We will evaluate the given integral.

Q: 4. If the curve y = In x between x = 1 and x = e is rotated about the y axis, which integral below…

A:

Q: Define the following improper integrals integral - infinity to 0  f(x) dx

A: Given: The improper integrals f(x)dx from limit -infinity to zero.

Q: Evaluate the double integral x cos y dA, where D is bounded by y = 0, y = x², and x = 5.

A:

Q: Evaluate the line integral, where C is the given curve, (x + 9y) dx + x2 dy,  C consists of line…

A: Consider the line integral,

Q: = Set up the definite integral required to find the area of the region between the graph of y y = 2x…

A: The integration formula for ∫xndx=xn+1n+1+C

Q: Guess the value of  integral 0 to infinity    x n e -x dx  , when is an arbitrarypositive integer.

A:

Q: In(x) d. f(x,y)dydx 1

A:

Q: 2 In y 1) Sketch the region R and express |e* dxdy as an equivalent double integral with the order…

A:

Q: Evaluate the triple integral ∫ 2 to 0 ∫ 2x to 0   ∫ ln x to 0 xe^−y dy dx dz

A: The given integral can be written as: I=∫20∫2x0∫lnx0xe-ydydxdz…

Q: Find the area above the curve y=Inx in the lower half-plane between x-0 and x=1. YA y=In x 0. O a. 1…

A:

Q: Find the value of the double integrals for: Ql: f(x, y) - ycos xy over the rectangle 0 sxS T, 0 sys1

A:

Q: compute the given approximation to the integral.

A:

Q: al is given below e integral over th yr²) dx.dy

A:

Q: Evaluate the integral. (Use C for the constant of integration.) dt t4 + 10

A:

Q: Set up the definite integral required to find the area of the region between the graph of y = 17 – r…

A:

Q: Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's rule to approximate the integral 13 In(x)…

A: (1) The trapezoidal rule gives: So, T12≈2.0810

Q: B: Evaluate the following integral by changing the order of integration: LS dy dx

A:

Q: b) Evaluate the following integrals using tabular integration by parts 1) (3x² – x + 2)e* dx +x + 1)…

A: Since you have posted multiple questions but according to guidelines, we will solve first question…

Q: consider the function y (x) = ln (x), from (1,0) to (3, ln (3)) b) Set up the integral that…

A: Setting up the integral

Q: Evaluate the integral. (Use C for the constant of integration.) t2 dt 1 - t6

A:

Q: Consider the function y (x) = 1 + ex / 2, from the point (0,2) to (4, e2). Set up the integral that…

A: Given that: y(x)=1+ex/2

Q: Find the area between the curves y=e−0.2xy=e-0.2x and y=1.6x+1y=1.6x+1 from x=0x=0 to x=4x=4.

A: Given query is to find the area under the curve

Q: over the graph y = e*, 0 < x < 1. Compute the path integral of f(x, y) = 2y² %D

A:

Q: 8. Find the area under the curve of f(t) = 1.2e0.1t betweent=1 and t = 2. %3D

A: From the given statement, the function

Q: Evaluate the double integral for R: 0<y<1,0<x<y² . Sly e*dxdy .

A:

Q: x3(1 + x4)5 dx,    u = 1 + x4

A:

Q: 1. Use Numerical methods to approximate the following integrals (a) (x +y)dxdy , n = m = 3

A: Trapezoidal rule: ∫cd∫abf(x, y)dxdy=h2fa, y+2fa+h, y+fa+2h, y+⋯+fa+n-1h, y+fb, ydy, where h=b-an.…

Q: Suppose a function y = f(x) is so be revolved at x = e from point (a,b) to (c, d), what is the…

A: The surface area generated by revolving the curve y = f(x) between x = a and x= b  revolved about x-…

Q: Evaluate the integral. (Use C for the constant of integration.) t5 In(t) dt

A:

Q: Approximate the integral fe* cos xdx using the composite Trapezoidal rule with M=5 subintervals. M-1…

A: The given integral is ∫01excos xdx. Approximate the integral ∫01excos xdx using the composite…

Q: which of the following integrals represents the area of the surface obtained by rotating the curve…

A:

Q: b. Find by double integration the area enclosed by the curve r= a(1+ cos 0) between 0 = 0 and 0=T.…

A:

Q: Set up the definite integral that gives the area of the shaded region bounded by the graphs of y,…

A:

Q: Evaluate the iterated integral. Triple Integral: (Outside to inside) Integral from 1 to 7, Integral…

A:

Q: - SxS Write an integral for the area of the surface generated by revolving the curve y = cos (3x)…

A:

Q: Evaluate the iterated integral below. π 1 S. *S'y cos (x) dy 0 0 dx

A: Here in above question we have to find iterated integral.

Q: Approximate the integral e-x dx as indicated below. Give your answer to 8 decimal places. Using the…

A:

Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.