Let the region R be the area enclosed by the function f (x) = 3x3 and g(x) = }x. x. Write an integral in terms of æ and also an integral in terms of y that would represent the area of the region R. If necessary, round limit values to the nearest thousandth. 6. 5 3 1 3 4 6 7 8 9 cx2 X2 dx x1 CY2 Yı = Y2 = |dy Y1

Q: x= 3+ y^2,x=2- y^2, y = 1, y = -2 O 7 sq. units O 8 sq. units O 9 sq. units 10 sq. units

A: We have to evaluate the area bounded by the curves x=3+y2 and x=2-y2 and the lines y=1 and y=-2. We…

Q: y = -x^2 + 4x, y = x^2 - 6x + 5, x=0, x= 1 O 2.012 sq. units O 2.022 sq. units O 2.032 sq. units…

A: We have to find the area of the bounded region.

Q: area is given by the function x^0.5+y^0.5=1

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

Q: y = -x^2 + 4x + 3, y = -x^3 + 7x^2 - 10x + 5, x = 1, x=2* 4.083 sq. units ○ 4.183 sq. units O 4.283…

A: We know that the area between the curves can be evaluated by double integration method. We will find…

Q: Let the region R be the area enclosed by the function f(x) = 2x³ , the horizontal line y = –3 and…

A: Given the region R be the area bounded by fx=2x13,y=-3,x=0,x=3. Visualize the solid: Each…

Q: Give an example showing how to use rectangles to estimate the area under a curve.  Then use…

A: Let us choose a function f(x) = x2 and say that we want to calculate the ara under f(x) and x-axis…

Q: Find the area bounded by y = x^2 -1, and the line y = 3. * 31/3 sq. units 35/3 sq. units O 37/3 sq.…

A:

Q: Determine the area, in square units, bounded above by f(x)=−x^2−10x−16 and g(x)=2x+16 and bounded…

A: Given that fx=-x2-10x-16gx=2x + 16 The objective is to find the area of the region which is bounded…

Q: Find the area under the curve y = x^2 on [3, 10]. a. Use four rectangles. b. Find the exact area.

A: a)-We have   fx= x2We want to calculate over the interval 3, 10 with 4 rectangles, thus…

Q: Find the points of intersection of the graphs of the functions y = (x − 4)³ and y = 6(x − 4) 4,8,0…

A: Step 1: To find the intersections of the functions, we will equate the two equations and solve for…

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.