When gas expands in a cylinder with radius , the pressureat any given time is a function of the volume:  P = PV.The force exerted by the gas on the piston (see the figure) isthe product of the pressure and the area:  F= pie r sq. P. Showthat the work done by the gas when the volume expandsfrom volume  V1 to volume V2 is W = integral V1 to  V2  P dV

Q: Find the area bounded by the following curves. y=x", x= -1, x = 2, y= 0 The area is (Simplify your…

A:

Q: This hemisphere has radius R=11 m; it is a water tank and water will enter or leave through the…

A: According to the question, To calculate the volume of water in hemisphere, V=23πR3=23π113=2787.64…

Q: The area bounded by the curve y=4- x2 · x-axis and the line is? 16 Ob. 11 Oc. 16 Od. 1 а. 5 3 3 3

A: The graph is shown below:

Q: The area of the shaded region is x3 + Vy = 1, 0<Isl Select one: a. 3 O b. 3 3 4 d. 9 14

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Q: Find the area under the curve y² = x and y=x a.) sq.units 2 b.) = sq.units

A: We need to find the area under the curve y2=x and y=x2. Let us first plot the region.

Q: At what rate is the area of rectangle changes if sides are x and 4x, if the shorter side x changes…

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Q: 15 m

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: An object moves along a line so that its velocity at time t is v(t)3t -30t+48 feet per second. Find…

A: Total distance travelled is integration of absolute value of (3t^2-30t+48) from -1 to 10.

Q: Find the area between the curves. x= 0, x= 3, y=2 eX, y= e³ 3x + e

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Q: Differenciate and simplify where possible

A: Introduction: Mathematicians use a procedure called differentiation to determine a function's…

Q: 10 2m | (100x – x³) dæ 16 V = = 27 blank – 104 = 27 (blank – = 27 (blank – 2500) =blank therefore…

A: Basic identity of integration is used

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.