8. Demonstrate the method of basic geometry to find the definite integrals. Include a rough sketch by hand..Rough pictures are acceptable, just be sure to convey the correct shape in order to show your work. (a) -2V49- 2 dr.

Q: Use Simpson's rule with 4 sub-intervals to find a rational approximation ton = L dx. 1+x2 a Give…

A: Solution:

Q: Find the integral. [col³ (x) csc ³¹ (x) dx (Use symbolic notation and fractions where needed. Use C…

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Q: d.x 4. x+ Va

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Q: 3. f(e* – x)dx

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Q: X (sinx 16)-()35in (ux) 13) 2020 +1 13

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Q: Q\ Use all your background to find the integral of the hyperbolic function: sinh ax -dx cosh bx

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Q: 3. A Estimate cos(z²) dz using a. the Trapezoidal Rule and b. the Midpoint Rule, each with n = 4.…

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Q: 2 Find the definite integral o tan(x)dx using strips of width for the trapezium rule correct to…

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Q: 5 x*-x³+3x² 1

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Q: 3 x dx 1

A: The given integral is ∫13xdx, with total n=4 subintervals. Comparing the given integral with…

Q: f. f S tan° r sec xdr tanx sec

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Q: V1 +x² – 1 In dx 8. xV1 +x? + C

A: Solving the given integral.

Q: Use the error formulas in Theorem errors in the Trapezoidal Rule and Simpson’s Rule to find n such…

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Q: 8 X COS X x2 − 5+6 -dx.

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Q: 5. Use basic area formulae from geometry to find the following definite integrals: - 2 if x 3. In…

A: Explanation of the answer is as follows

Q: Give an approximation of the following defined integral with an absolute error below 10-8 3 :/ Use…

A: Given integral is I=∫13x2x3−e−xdx. We have to find the approximate value of the given integral.…

Q: consider the function y=f(x)= x + sin(x).  What is the integral formula for the length of the graph…

A: The problem concerns determination of arc length along the given curve (using integration)

Q: How many evaluations of the function f(x) = (1+ 2x)¬1 is needed in order to approxi- mate the…

A: Given: fx=1+2x-1   To determine: The number of evaluations of the given function needed to…

Q: Evaluate the integrals. It may be necessary to use a substitution first. ∫ ds/es - 1

A: consider the given integral as

Q: Set up the integral used to find the length of the curve  y = sqrt(1 − x^5 on the interval -(1/4)…

A: The length of curve y=f(x) from x=a to x=b is L=∫ab1+dydx2dx Given function is y=f(x)=1-x5…

Q: How does the method of substitution work for definite integrals? Give examples.

A: To elaborate the use of substitution for computing the integrals with example.

Q: state the trigon metri e subs tition you would use to Find the indefinite integral Do Not integrate.…

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Q: a) Determine whether the equation is exact. If it is exact, then: (b) Solve the equation.  (c) Use…

A: Given differential equation dydx=-4x+2y2x+3y To find: (a) To determine whether the equation is exact…

Q: 1 dæ with n = 4. Use the trapezoid rule and Simpson's rule to approximate the value of (7+ x³ Note…

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Q: Use the Change of Variables Formula to evaluate the definite integral. 1 dx V3x + 5 (Use symbolic…

A: Given:       ∫0413x+5dx      substitute u=3x+5→dudx=3→dx=13du            =∫0413udu taking constant…

Q: V25 – 16x 2. 5-V25 16x2 dx% 3D51n + / 25 - 16x2 +C 4x

A: we have to find the following integral

Q: Use any method to evaluate the integrals. Mostwill require trigonometric substitutions, but some can…

A: To evaluate the given integral

Q: 3. Evaluate the definite integral.

A: Given, ∫121t+tdt We have to evaluate this indefinite integral

Q: Use the Midpoint Rule with n = 4 to approximate the value of the definite integral. Use a graphing…

A: fx=secx,  0, 1 Now, ∆x=b-an=1-04=14 Now the subintervals are 0, 14,  14, 12,  12, 34  &  34, 1.…

Q: Problém 2: For each part, (i) write a definite integral that computes the arc length of the curve…

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Q: Q2M) A tank has the shape of an inverted right pyramid with a square base, as shown in the picture.…

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Q: Use the definite integral and the Fundamental Theorem of Calculus to solve the following problem. A…

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Q: Use the trapezoidal rule and Simpson's rule to approximate the value of the definite integral. (Give…

A: Given:

Q: 1. Using the geometric interpretation of the definite integral, find the value of the integral be So…

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Q: Set up the integral to find the length of the curve: f(x) = sqrt(4-x^2), -2<= x <=1. Do not…

A: Given the function is

Q: Use the error formulas to find bounds for the error in approximating the definite integral using the…

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Q: - x2 + 9x between 6. In your calculations, if you need to round, do not do so until the very end of…

A: Area under curve = integration of function in respective given limits

Q: (a) Use the Midpoint Rule with n = 6 to approximate integral Sc dx 8 x-6 x+8 Write your answer exact…

A: (a) From the given integral, the function is fx=x-6x+8, a=6, b=8 and n=6. Note that, the formula for…

Q: dx 1 (Vx² + 36 – 6' -In + C xVx2 + 36 6

A: Given

Q: Use Simpson's rule (n = 4-this means that there are 2 rectangles for the trapezoid rule and 2…

A: To evaluate ∫-11x2-1 dx with n=4 using Simpsons Rule

Q: Find the area of the triangle formed by the functions using definite integrals f(x) = 2x+4 g(x) =…

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Q: Use the dennition of the Integral to evaluate x2 dx Be sure to carefully justify all of your steps.…

A: Simple use the given identity i the solution page go Through that.. And finally put the upper and…

Q: Find the exact value of the integral using formulas from geor 5. (2+2x)dx 2. |(2 + 2x)dx = (Simplify…

A: Given: ∫25(2+2x)dx To find the exact value of the integral using geometry

Q: 3. Use properties of definite integrals and appropriate formulas from geometry to evaluate the…

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Q: x² sin xdx 2. 0,

A: See the details solution in below

Q: 2. Using f adr = and fz*dr = . 3, use properties of the definite integrals to find (1 + 21)dr.

A: We have to find the integral of given function.

Q: 3 Find approximation a5 costect to an three decimal places.

A: We approximate the value by using differential. The answer is given as follows

Q: cot 3x csc 3xdx

A: Consider the given integral.

Q: 2. Given y =-x³ - 6x2 +15x, find inf lection point s, the relative extrema, int ervals where the…

A: y=-x3-6x2+15x first find the critical point of given function  So find derivative now  y' =…

Q: Set up an integral that represents the length of the curve. Then use your calculator to find the…

A: We will not solve this for y. Instead, we are going to use the formula  We find dx/dy first

Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus. 

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.