Help to Integral Calculus Solutions & Examples

Give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus.
Find a polar equation for each conic section. ${\displaystyle e=1,y=2}$

Evaluate the indefinite integral.
${\displaystyle \int \left(\csc t\cot t\right)dt}$

Change the following decimals to fractions. Give each fraction in its simplest form.
a) 4.2,
b) 0.06,
c) 1.85,
d) 2.005.

Explain why each of the following integrals is improper.
(a) ${\int }_6^7\frac{x}{x-6}dx$
-Since the integral has an infinite interval of integration, it is a Type 1 improper integral.
-Since the integral has an infinite discontinuity, it is a Type 2 improper integral.
-The integral is a proper integral.
(b)${\int }_0^{\mathrm{\infty }}\frac{1}{1+x^3}dx$
Since the integral has an infinite interval of integration, it is a Type 1 improper integral.
Since the integral has an infinite discontinuity, it is a Type 2 improper integral.
The integral is a proper integral.
(c) ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}^2{e}^{-x^2}dx$
-Since the integral has an infinite interval of integration, it is a Type 1 improper integral.
-Since the integral has an infinite discontinuity, it is a Type 2 improper integral.
-The integral is a proper integral.
d)${\int }_0^{\frac{\pi }{4}}\cot xdx$
-Since the integral has an infinite interval of integration, it is a Type 1 improper integral.
-Since the integral has an infinite discontinuity, it is a Type 2 improper integral.
-The integral is a proper integral.

Write the decimals in words. Write the money amounts in words for dollars and fractions for cents. Write the words in decimals. 0.4

Use a double integral to find the area of the region. The region inside the cardioid $r=1+\cos \theta$ and outside the circle $r=3\cos \theta$

Change from rectangular to cylindrical coordinates. (Let $r\ge 0$ and $0\le \theta \le 2\pi$.)
a) $(-2,2,2)$
b) $(-9,9\sqrt{3,6})$
c) Use cylindrical coordinates.
Evaluate
$\int \int {\int }_{E}xdV$
where E is enclosed by the planes $z=0$ and
$z=x+y+10$
and by the cylinders
$x^2+y^2=16$ and $x^2+y^2=36$
d) Use cylindrical coordinates.
Find the volume of the solid that is enclosed by the cone
$z=\sqrt{x^2+y^2}$
and the sphere
$x^2+y^2+z^2=8$.

Determine whether the given vectors are orthogonal, parallel,or neither:
$u=(-3,9,6)$
$v=(4,-12,-8)$

Use the table of values of $f(x,y)$ to estimate the values of $fx(3,2)$, $fx(3,2.2)$, and $fxy(3,2)$.
$\begin{array}{|cccc|}\hline y& 1.8& 2.0& 2.2\\ x& & & \\ 2.5& 12.5& 10.2& 9.3\\ 3.0& 18.1& 17.5& 15.9\\ 3.5& 20.0& 22.4& 26.1\\ \hline\end{array}$

1) Find the area of the part of the plane
$4x+3y+z=12$
that lies in the first octant.
2) Use polar coordinates to find the volume of the given solid.
Bounded by the paraboloid $z=5+2x^2+2y^2$ and the plane z = 11 in the first octant

Show that the decimals 0.6, 0.1, 0.2, and 0.325 are positive rational numbers.

Evaluate this indefinite integral: $\int (3x+5)dx$

Express as a trigonometric function of one angle.
a) $\cos 2\sin (-9)-\cos 9\sin 2$
Find the exact value of the expression.
b) $\sin (\arcsin \frac{\sqrt{3}}{2}+\arccos 0)$

Describe in words the surface whose equation is given. $\varphi =\frac{\pi }{4}$ (select the correct answer)
1)the top half of the right circular cone with vertex at the origin and axis the positive z-axis
2)the plane perpendicular to the xz-plane passing through z = x, where $x\ge 0$
3)the plane perpendicular to the xy-plane passing through y = x, where $x\ge 0$
4)the base of the right circular cone with vertex at the origin and axis the positive z-axis
5)the plane perpendicular to the yz-plane passing through z = y, where $y\ge 0$

Find the length of the curve.
$r(t)=⟨6t,t^2,\frac{1}{9}{t}^3⟩,0\le t\le 1$

True or False: A basis of R2[x] can be extended to a basis of R3[x].

Evaluate the integral by reversing the order of integration
${\int }_0^1{\int }_{3y}^3{e}^{x^2}dxdy$

Given ${\int }_2^{5}f(x)dx=17$ and ${\int }_2^{5}g(x)dx=-2$, evaluate the following.
(a)${\int }_2^5[f(x)+g(x)]dx$
(b)${\int }_2^5[g(x)-f(x)]dx$
(c)${\int }_2^{5}2g(x)dx$
(d)${\int }_2^{5}3f(x)dx$

Find an equation of the plane tangent to the following surface at the given point.
$7xy+yz+4xz-48=0;(2,2,2)$