Find the area between the curve y = x² + 7x + 12 and the x-axis from x = -3 to x = -1. Step 1 Notice that the curve is a parabola with x-intercepts 0) and (-4, 0). Within our interval [-3, –1], f(x) is -Select---

Q: 1) Find the area of the region above the x axis and to the left of the line x = 1 bounded by the x…

A:

Q: Q2/ Find the area of a region bounded by the curve f(x) = 8x' and x-axis and the two lines x=1, x=3.…

A:

Q: Find the area under the curve 4x²-3x+6 between x = -2 and x=3

A:

Q: Find the area beneath the curve g(x) = 8x + 3x2 from x = 5 to x = 6

A: Given function is,

Q: Q4. Find the area between curve y = x - 3x + 2 and x-axis from x =0 to x 2

A:

Q: 6. Find the area between the curve y = x(x - 3) and the ordinates x = 0 andx = 5.

A:

Q: Find the area between the curves x² and 3x – 2

A:

Q: 3. Find the exact area between the curve f (x) = -2x² + 12x + 7, the x-axis, and x=1 and x=5.

A: As mentioned, use the sigma formula to find the area A. Plot the curves. Divide the area between the…

Q: Find the area under the curve y = x3 + 3x2 and the x-axis between x = 1, and x = 3

A: Find the area under the curve y = x3 + 3x2 and the x-axis between x = 1, and x = 3.

Q: 5. Find the area between the curve y, =x² - 4x +12 and y, = x² from 0<x<4

A: Given curves are y1=x2-4x+12y2=x2 Also 0≤x≤4

Q: Find the area between the curves f(x) = −x² + 4x and g(x) = x² - 6x + 8.

A: Basic identities of integration are used

Q: Find the area of the region bounded by the parabola y = 4x², the tangent line to this parabola at…

A:

Q: 1) Find the area of the region above the x axis and to the left of the line x = 1 bounded by the x…

A:

Q: Find the total area between the curve of the function f(z) = -4z +8 and the z-axis over the interval…

A: I have provided solution in  step2.

Q: 1. Find the area in the second quadrant bounded by the curve 2x² + 4x +y = 0 using two methods

A:

Q: Find the area of the region bounded by the parabola y = 2x, the tangent line to this parabola at (4,…

A: Definition used -              Equation of a tangent line -                  (y-y1)=f'(x1)(x-x1)…

Q: B- Find the area between the curve (y-√x - 2) and the lines (y-4-x),(y=0).

A: We have given three curves-y=x-2y=4-xy=0 First we will find intersection of first two curves…

Q: 1. Find the area between the curves x2-y = 9 and the line y 2x-6

A: Since you have posted multiple questions in this question which are not interlinked so I do only…

Q: 1. Find the area between the curves x2-y2 = 9 and the line y = 2x-6

A:

Q: b) Find the area between the 2 curves f(x)=x²+4-4x and x-axes and border with the two lines = 5 &x=…

A:

Q: find the total area between the curve g(x)=x^3 + 2x^2 -x-2 and the x-axis, from x=0 to x=2

A:

Q: 1 Find the area bounded by the curve y=- =-=-—-—-x³-3³² x ² + 16 4 -x2 + 3x and the line y=x.

A:

Q: The area under the curve y = 2x bounded from below bythe x-axis and on the right by x = 4 is 16 3…

A: Given that the area under the curve y=2xwe need find the area below the x-axis and the right by x=4

Q: Find the area of the “triangular” region in the first quadrant that is bounded above by the curve y…

A: Consider the provided curve,  Bounded above by the curve y=ex2 and below by the curve y=e-x2. And on…

Q: Find the area bounded by the loop of the curve y² = 4x²-x*

A: The given equation of the curve is: This can also be written as: The graph of this curve is:

Q: Find the area of the region bounded by the parabola y = 4x2, the tangent line to this parabola at…

A: The equation of the parabola is given by y = 4x2.Differentiate y = 4x2 with respect to x as follows.

Q: Find the area of the region bounded by the parabola f(x) = 16-x^2 and the lines  x=0 and x = 4 and…

A: Givenf(x) = 16-x2lines x=0, x=4 and x-axis(y=20)

Q: 2. Find the total area under the curve f(x) = 2xe² from x = 0 and x = 5.

A:

Q: The area between the curve y = = and the x-axis from x = 10 to x = 20 is the same as the area…

A:

Q: 2 Find the area of a triangle bounded by the y axis, the line f(x) = 8 – -x, and the line…

A:

Q: Find the area between the curve ​f(x)=x^2​+15 and the​ x-axis from x=3 to x=5.

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

Q: - Determine the area bounded by the curves y = -1² – 16x + 80 and y = (x – 4)°.

A:

Q: 7. Find the area bounded by the curve y = x exp(-x²), the x-axis and the vertical line p through the…

A:

Q: The area bounded by the curves: у-х, у-6-х аnd у-0.5х is:

A: see below the answer and explanation

Q: - 5. Find the area bounded by the curve y² − 3x + 3 = 0 and the line x = 4

A:

Q: 3. Find the area bounded by the curve y = x² - 7x + 6 and the x-axis and the ordinates x 2 and x =…

A:

Q: Find the area between the curves. y = x2 - 9x + 4 and y = x² + 6 from x = 0 to x = 6

A:

Q: 6. Find the area bounded by the curves of y = 6x + x² andy = -x² - 2x, and the linex = 5.

A: The equation of the curves are  C1: y = 6x+x2C2: y = -x2-2xC3: x = 5 To evaluate: The area of the…

Q: 11. Find the area bounded by the curve y² + 2x - 2y - 3 = 0 and the y-axis.

A:

Q: Find the area of the region bounded by the parabola y = 4x2, the tangent line to this parabola at…

A:

Q: Find the area under the curve 4x2-3x+6 between x= -2 and x=3

A:

Q: Find the area of the region between the curves y=x2-1 and y= - 2x² +2 from x= - 1 to x=1.

A:

Q: 2 Find the area between the curves y= 2x +6x+1 and y =-x +3x+7 over the interval [-4,10]. Draw the…

A:

Q: Find the area between the curve f(x) = −3x^2 + 3 and the x-axis over the interval [−1, 2].

A: Answer and explanation is given below

Q: 14. (a) Find the area bounded by the curves y = 2x² and y = 4x. %3D (b) Two cars both start on a…

A:

Q: Find the area of a triangle bounded by the y axis, the line f(x) = 9 -x, and the line perpendicular…

A:

Q: 3 2. Find the area bounded by the two curves y = x³- 6x² + 8x and y = x² - 4x.

A:

Q: Find the area of a triangle bounded by the yy-axis, the line f(x)=10−2xf(x)=10−2x, and the line…

A: Now, from the graph we have the points of intersection of the lines

Q: Find the area bounded by the curve f(x) = x³ + 3x² – 4x – 5 and x-axis from x = -3 to x = -1 using:…

A: Consider the given area bounded by the curves as y=x3+3x2-4x-5 and the x axis from x=-3 to x=-1. Let…

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.