OEF geometric integral --- Introduction ---

This module actually contains 19 exercises on geometric applications of definite integrals of one variable: area, barycenter, arc length, etc.

There are other modules of exercises on definite integrals: OEF definite integral for theory and computation of definite integrals, and OEF physical integral for applications on physics.

Arc length explicit 2D

Compute the length of the curve , between and .

xrange , yrange , arrow ,0,,0,10,grey arrow 0,,0,,10,grey plot skyblue, trange , linewidth 3 plot blue,

Please give your reply with a precision of at least 4 digits after the decimal point.

Arc length parametric 2D

Compute the length of the parametric curve

,

between and .

xrange , yrange , trange , arrow ,0,,0,10,grey arrow 0,,0,,10,grey plot skyblue,, trange , linewidth 3 plot blue,,

Please give your reply with a precision of at least 4 digits after the decimal point.

Crossed cubic area

Compute the area of the blue region below, where the red curve is that of the function f(x) = .

Please give your result of computation with a precision of at least 5 decimal places.

Circular distance

A point turns around a circle of radius with constant speed, and another point stays at a constant position at distance of towards the center of the circle. Compute the average distance between the two points.

Please give your reply with a precision of at least 4 digits after the decimal point.

Cubic area

Compute the area of the shaded region below, where C is the curve of the function f(x) = 3-, and L is a horizontal line tangent to C.

Please give your result of computation with a precision of at least 5 decimal places.

Eclipse area

Here we have a partial sun eclipse, where the shade of the moon has a radius exactly equal to that of the sun, and the distance between the centers of the sun and the moon equals times the radius of the sun. Compute the percentage of the eclipse, that is, the percentage of the sun surface (as a disk) hidden by the moon.

Please give the result of your computation with a precision of 0.1% or better.

Polar length closed

The following curve is defined by the polar equation , where is the polar angle. Compute the length of the marked part of this curve.

xrange -, yrange -, fill 0,0,white tstep 500 trange , arrow 0,0,,0,12,grey disk 0,0,6,red plot skyblue,()*cos(t),()*sin(t) trange , linewidth 2 plot blue,()*cos(t),()*sin(t)

Polar length open

The following curve is defined by the polar equation , where is the polar angle. Compute the length of this curve for going from to .

xrange -, yrange -, fill 0,0,white tstep 500 trange , arrow 0,0,,0,12,grey disk 0,0,6,red plot skyblue,()*cos(t),()*sin(t) trange , linewidth 2 plot blue,()*cos(t),()*sin(t)

Polar length spiral

The following curve is defined by the polar equation , where is the polar angle. Compute the length of this curve for going from to .

xrange -, yrange -, fill 0,0,white tstep 500 trange , arrow 0,0,,0,12,grey disk 0,0,6,red plot skyblue,()*cos(t),()*sin(t) trange , linewidth 2 plot blue,()*cos(t),()*sin(t)

Given log area

Consider the function . The following picture shows the curve of f (x). The red vertical line in the picture is given by an equation x=c. Given that the blue region has an area equal to , what is the value of c?

xrange , yrange , arrow ,0,,0,10,grey arrow 0,,0,,10,grey text grey,0.95*,0.1*,small,x text grey,0.03*,0.98*,small,y trange 0, plot black,t, vline ,0,red fill 0.8*,0.2*,skyblue

Please give your reply with a precision of at least 4 digits after the decimal point.

Parabolic barycenter

Compute the barycenter p0=(x0,y0) of the shaded region below, where C is the curve of the function f(x) = .

Please give your result of computation with a precision of at least 5 decimal places.

Parabolic area

Compute the area of the shaded region below, where C is the curve of the function f(x) = 2, and L is the line defined by the equation +=.

Please give your result of computation with a precision of at least 5 decimal places.

Parabolic area II

Compute the area of the shaded region below, where C is the curve of the function f(x) = 2, and the two lines L1 and L2 are given by y= and y= respectively.

Please give your result of computation with a precision of at least 5 decimal places.

Parabole+circle area

Compute the area of the shaded region below, where C is a circle of radius with center at the origin, and P is the curve of the function f(x) = .

Please give your result of computation with a precision of at least 5 decimal places.

Crossed quadratic area

Compute the area of the blue region below, where C is the graph of the function f(x) = , and L is the line x = .

Please give your result of computation with a precision of at least 5 decimal places.

Given quadratic area *

Consider the function . Its curve is shown by the following picture. Given that the area of the yellow region is equal to , what is the value of c?

xrange , yrange , plot black, arrow ,0,,0,10,grey fill ,*(-1)*0.1,yellow arrow 0,,0,,10,grey text grey,-0.04*,0.07*,small,x text grey,0.03*,-0.01*,small,y

Please give your reply with a precision of at least 2 digits after the decimal point.

Spherical reservoir

A factory has a water reservoir under the form of a ball of meters of (internal) diameter. The usual water level is of meters above the bottom of the reservoir. One day, due to a system breakdown, this level has dropped to meters. How many cubic meters of water should be pumped into the reservoir, in order to get it back to the usual level?

Surface of revolution X

Compute the surface of the solid du solide resulting from the rotation of the following red curve

round the axis, for going from to .

xrange , yrange , arrow ,0,,0,10,grey arrow 0,,0,,10,grey text black,-0.04*(),0.08*(),small,x text black,0.03*(),-0.02*(),small,y dline ,0,,,black dline ,0,,,black trange , plot red, v=0.3 u=0.8 r=-0.12*() m=0.03* n=0.07* trange v,2*pi-v plot black,m*cos(t)+r,n*sin(t) arrow m*cos(u)+r,n*sin(u),m*cos(v)+r,n*sin(v),8,black

Volume of revolution X

Compute the volume of the solid resulting from the rotation of the following red curve

around the axis, for going from to .

xrange , yrange , arrow ,0,,0,10,grey arrow 0,,0,,10,grey text black,-0.04*(),0.08*(),small,x text black,0.03*(),-0.02*(),small,y dline ,0,,,black dline ,0,,,black trange , plot red, v=0.3 u=0.8 r=-0.12*() m=0.03* n=0.07* trange v,2*pi-v plot black,m*cos(t)+r,n*sin(t) arrow m*cos(u)+r,n*sin(u),m*cos(v)+r,n*sin(v),8,black
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