# Exercise

Find the limits

1. (Put . Show that as )

2. (). []

3. . []

4. (Let Consider ) []

5. ().

# Exercise

1. Find the area of the region situated in the first quadrant and bounded by the parabola and the straight lines and

2. Find the area of the region bounded by the curve and the arc of the circle and situated outside the circle. []

3. Compute the area of the figure bounded by the circle and

4. Find the area of the figure cut out by the circle from the cardioid

5. Find the area of the region enclosed by the loop of the folium of Descartes by converting to polar coordinates. []

6. Compute the area of the figure bounded by the straight lines and the curves

7. Compute the area of the figure bounded by the parabolas

8. Find the area of the figure contained between the parabola and the witch of Agnesi

9. Find the area of the figure which lies in the first quadrant inside the circle and is bounded by the parabolas and

10. Compute the area of the figure bounded by the lines and the x-axis.[]

11. Find the area of the segment of the curve if the line is the chord determining the segment.[]

12. Compute the area enclosed by the loop of the curve

13. Find the area of the figure bounded by the parabola the line tangent to it at the point and the y-axis.[]

14. Find the area bounded by the parabolas and the straight line

15. Find the area enclosed by the circle and the parabola

If the boundary of a figure is represented by parametric equations

then the area of the figure is evaluated by

1. Find the area enclosed by the ellipse

2. Find the area enclosed by the astroid

3. Find the area of the region bounded by an arc of the cycloid and the x-axis. []

4. Compute the area of the region enclosed by the curve

5. Find the area of the region enclosed by the loop of the curve

6. Find the area enclosed by the cardioid

The arc length of a curve in polar coordinates

If a smooth curve is given by the equation in polar coordinates, then the arc length of the curve is expressed by the integral:

where and are the values of the polar angle at the endpoints of the arc ().

1. Find the length of the first turn of the spiral of Archimedes []

2. Find the arc length of the cardioid []

3. Find the length of the closed curve

4. Find the length of the curve between and .[]

Area of surface of revolution

The area of the surface generated by revolving about the x-axis the arc of the curve is expressed by the integral

1. Find the area of the surface formed by revolving the astroid

2. Find the area of the surface generated by revolving about the x-axis of a closed contour formed by the curves and

3. Find the area of the surface obtained by revolving a loop of the curve about the y-axis.

4. Compute the area of a surface generated by revolving about the x-axis an arc of the curve between the points of intersection of the curve and the x-axis. []

5. Compute the surface area of the torus generated by revolving the circle about the x-axis.

6. Compute the area of the surface formed by revolving the lemniscate about the polar axis.

7. Compute the area of the surface formed by revolving one branch of the lemniscate about the straight line

# Exercise

The volume of a solid is expressed by the integral

where is the area of the section of the solid by a plane perpendicular to the x-axis at the point with coordinate and are the left and right boundary of variation of

1. Find the volume of the ellipsoid

2. The axes of two identical cylinders with bases of radius intersect at right angles. Find the volume of the solid constituting the common portion of the two cylinders.[]

3. On all chords parallel to one and the same direction of a circle of radius symmetrical parabolic segments of the same altitude are constructed. The planes of the segments are perpendicular to the plane of the circles. Find the volume of the solid thus obtained.[]

4. Compute the volume of the solid generated by revolving about the x-axis the area bounded by the axes of coordinates and the parabola

5. The figure bounded by an arc of the sinusoid , the y-axis and the straight line revolves about the y-axis. Compute the volume of the solid of revolution thus generated.[]

6. Compute the volume of the solid generated by revolving about the x-axis the figure bounded by the parabola and the straight line

7. Compute the volume of the solid generated by revolving about the y-axis the figure bounded by the parabolas and

8. Find the volume of the solid generated by revolving about the line the figure bounded by the parabola and the straight line

9. Find the volume of the solid generated by revolving about the x-axis the figure enclosed by the astroid

10. Compute the volume of the solid obtained by revolving about the x-axis the cardioid

Computing limits of sums with the aid of definite integrals

If is a continuous function defined on , then the Riemann sum

11. Compute

12. Compute

13. Compute

Average Value of a Function

The average value of over the interval [a,b] is the number

1. Find the average value of the function over the interval

2. Find the average length of all vertical chords of the hyperbola over the interval

3. Find the average value of the function over the interval

This document created by Scientific WorkPlace 4.0.