Find the limits
(Put . Show that as )
(). []
. []
(Let Consider ) []
().
Find the area of the region situated in the first quadrant and bounded by the parabola and the straight lines and
Find the area of the region bounded by the curve and the arc of the circle and situated outside the circle. []
Compute the area of the figure bounded by the circle and
Find the area of the figure cut out by the circle from the cardioid
Find the area of the region enclosed by the loop of the folium of Descartes by converting to polar coordinates. []
Compute the area of the figure bounded by the straight lines and the curves
Compute the area of the figure bounded by the parabolas
Find the area of the figure contained between the parabola and the witch of Agnesi
Find the area of the figure which lies in the first quadrant inside the circle and is bounded by the parabolas and
Compute the area of the figure bounded by the lines and the x-axis.[]
Find the area of the segment of the curve if the line is the chord determining the segment.[]
Compute the area enclosed by the loop of the curve
Find the area of the figure bounded by the parabola the line tangent to it at the point and the y-axis.[]
Find the area bounded by the parabolas and the straight line
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
Find the area enclosed by the ellipse
Find the area enclosed by the astroid
Find the area of the region bounded by an arc of the cycloid and the x-axis. []
Compute the area of the region enclosed by the curve
Find the area of the region enclosed by the loop of the curve
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
().
Find the length of the first turn of the spiral of Archimedes []
Find the arc length of the cardioid []
Find the length of the closed curve
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
Find the area of the surface formed by revolving the astroid
about the x-axis.
[]
Find the area of the surface generated by revolving about the x-axis of a closed contour formed by the curves and
Find the area of the surface obtained by revolving a loop of the curve about the y-axis.
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. []
Compute the surface area of the torus generated by revolving the circle about the x-axis.
Compute the area of the surface formed by revolving the lemniscate about the polar axis.
Compute the area of the surface formed by revolving one branch of the lemniscate about the straight line
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
Find the volume of the ellipsoid
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.[]
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.[]
Compute the volume of the solid generated by revolving about the x-axis the area bounded by the axes of coordinates and the parabola
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.[]
Compute the volume of the solid generated by revolving about the x-axis the figure bounded by the parabola and the straight line
Compute the volume of the solid generated by revolving about the y-axis the figure bounded by the parabolas and
Find the volume of the solid generated by revolving about the line the figure bounded by the parabola and the straight line
Find the volume of the solid generated by revolving about the x-axis the figure enclosed by the astroid
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
Compute
Compute
Compute
Average Value of a Function
The average value of
over the interval [a,b] is the number
Find the average value of the function over the interval
Find the average length of all vertical chords of the hyperbola over the interval
Find the average value of the function over the interval