Exercise

Find the limits

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  10. MATH (Put $26+x=z^3$. Show that $z\rightarrow 3$ as $x\rightarrow 1.$)$[54]$

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  62. MATH ($a>0$). $[\ln a]$

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Exercise

  1. Find the area of the region situated in the first quadrant and bounded by the parabola $y^{2}=4ax$ and the straight lines $y=x-a$ and $x=a.$ MATH

  2. Find the area of the region bounded by the curve $r=2a\cos 3\theta $ and the arc of the circle $r=a$ and situated outside the circle. [MATH]

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

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

  5. Find the area of the region enclosed by the loop of the folium of Descartes $x^{3}+y^{3}=3axy$ by converting to polar coordinates. [$\frac{3}{2}a^{2}$]

  6. Compute the area of the figure bounded by the straight lines $x=0,x=2$ and the curves MATH

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

  8. Find the area of the figure contained between the parabola $x^{2}=4y$ and the witch of Agnesi MATH

  9. Find the area of the figure which lies in the first quadrant inside the circle $x^{2}+y^{2}=3a^{2}$ and is bounded by the parabolas $x^{2}=2ay$ and $y=2ax$ MATH

  10. Compute the area of the figure bounded by the lines $y=x+1,y=\cos x$ and the x-axis.[$\frac{3}{2}$]

  11. Find the area of the segment of the curve $y^{2}=x^{3}-x^{2}$ if the line $x=2$ is the chord determining the segment.[$\frac{32}{15}$]

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

  13. Find the area of the figure bounded by the parabola $y=-x^{2}-2x+3,$ the line tangent to it at the point $(2,-5)$ and the y-axis.[$\frac{8}{3}$]

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

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

If the boundary of a figure is represented by parametric equations
MATH
then the area of the figure is evaluated by
MATH

  1. Find the area enclosed by the ellipse MATH $[\pi ab]$

  2. Find the area enclosed by the astroid MATH

  3. Find the area of the region bounded by an arc of the cycloid MATH and the x-axis. [$3\pi a^{2}$]

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

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

  6. Find the area enclosed by the cardioid MATH

The arc length of a curve in polar coordinates

If a smooth curve is given by the equation $r=r(\theta )$ in polar coordinates, then the arc length of the curve is expressed by the integral:
MATH
where $\theta _1$ and $\theta _2$ are the values of the polar angle $\theta $ at the endpoints of the arc (MATH).

  1. Find the length of the first turn of the spiral of Archimedes $r=a\theta .$ [MATH]

  2. Find the arc length of the cardioid MATH [$8a$]

  3. Find the length of the closed curve MATH

  4. Find the length of the curve MATH between $r=2$ and $r=4$.[$3+\frac{\ln 2}{2}$]

Area of surface of revolution

The area of the surface generated by revolving about the x-axis the arc $L$ of the curve $y=y(x),a\le x\le b$ is expressed by the integral
MATH

  1. Find the area of the surface formed by revolving the astroid
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    about the x-axis. [MATH]

  2. Find the area of the surface generated by revolving about the x-axis of a closed contour formed by the curves $y=x^{2}$ and MATH

  3. Find the area of the surface obtained by revolving a loop of the curve MATH about the y-axis.$[3\pi a^{2}]$

  4. Compute the area of a surface generated by revolving about the x-axis an arc of the curve MATH between the points of intersection of the curve and the x-axis. [$3\pi $]

  5. Compute the surface area of the torus generated by revolving the circle MATH about the x-axis. $[4\pi ^{2}br]$

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

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

Exercise

The volume of a solid is expressed by the integral
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where $S(x)$ is the area of the section of the solid by a plane perpendicular to the x-axis at the point with coordinate $x;a$ and $b$ are the left and right boundary of variation of $x.$

  1. Find the volume of the ellipsoidMATH

  2. The axes of two identical cylinders with bases of radius $a$ intersect at right angles. Find the volume of the solid constituting the common portion of the two cylinders.[$\frac{16}{3}a^{3}$]

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

  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 MATH

  5. The figure bounded by an arc of the sinusoid $y=\sin x$, the y-axis and the straight line $y=1$ revolves about the y-axis. Compute the volume $V$ of the solid of revolution thus generated.[MATH]

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

  7. Compute the volume of the solid generated by revolving about the y-axis the figure bounded by the parabolas $y=x^{2}$ and MATH

  8. Find the volume of the solid generated by revolving about the line $y=-2a$ the figure bounded by the parabola $y^{2}=4ax$ and the straight line MATH

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

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

    Computing limits of sums with the aid of definite integrals

    If $f$ is a continuous function defined on $[0,1]$, then the Riemann sum
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  11. Compute MATH

  12. Compute MATH

  13. Compute MATH

Average Value of a Function

The average value of $f(x)$ over the interval [a,b] is the number
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  1. Find the average value of the function MATH over the interval MATH

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

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

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