Integral Problem Solving Made Easy: Learn with Examples

Provided a function that would be defined fo every natural number n as: $$g(n)=\frac{2}{2}+\frac{1}{4}+\frac{1}{8}+\cdot \cdot \cdot +\frac{1}{2^{n-1}}+\frac{1}{2^n}$$ COnjecture a closed form form for g(n) (a simple expression devoid of signs or dots) while also using mathematical induction to prove that the conjecture is true.

Evaluate integration
${\displaystyle \int \frac{1}{{(x^2-9)}^{\frac{3}{2}}}dx}$
using trigonometry substitution

Change from rectangular to cylindrical coordinates. (Let ${\displaystyle r\ge 0\text{and}0\le \theta \le 2\pi }$.)

(A) (−2, 2, 2)

(B) ${\displaystyle (-9,9\sqrt{3},6)}$

(C) Use cylindrical coordinates. Evaluate
${\displaystyle \int {\int }_E\int xdV}$
where E is enclosed by the planes z = 0 and
${\displaystyle z=x+y+10}$
and by the cylinders ${\displaystyle x^2+y^2=16\text{and}{x}^2+y^2=36.}$
D) Use cylindrical coordinates. Find the volume of the solid that is enclosed by the cone
${\displaystyle z=\sqrt{x^2+y^2}}$
and the sphere
${\displaystyle x^2+y^2+z^2=8.}$

Integrate $\int \arctan (2\theta )d\theta$

Find ${\displaystyle \int 2x^3\cos \left(x^2\right)dx}$

${\displaystyle \int 6x^2{(x^3+2)}^{99}dx}$

${\displaystyle \int \frac{\sqrt{x^2-1}}{x^4}dx}$

Evaluate the following problem.
${\displaystyle \int x^3{(a^2+x^2)}^{\frac{1}{2}}dx}$

Calculate the integral ${\displaystyle \oint \frac{dz}{e^{z-1}}}$ over the circle of radius r=4 centered at z=4i

Pollen analysis, or palynology, is an important tool used in archaeology. Palynologists take samples of soil from archaeological sites and analyze the pollen from different soil layers. By examining the changes in pollen types over time, palynologists can learn about historical land use. The pollen in the soil indicates how the land was used—whether it was cultivated, a forest was cleared, or if it was abandoned. Scientists have been trying to find the origin of corn. They know that corn was domesticated from a plant called teosinte that grew somewhere in the central valley of Mexico between 12,000 and 6,000 years ago. It often is hard to find intact corncobs because they do not fossilize well. How could a palynologist help determine the origin of corn?

Mass Analysis of Two Chlorine-Fluorine Samples SampleMass of Chlorine (g)Mass of Fluorine (g)% Cl% F I13.0226.97865.1134.89 II5.7539.248??
Which statement best describes the relationship between the two samples?
A. The compound in Sample I is the same as in Sample II. Therefore, the mass ratio of Cl to F in both samples will obey the law of definite proportions.
B. The compound in Sample I is the same as in Sample II. Therefore, the mass ratio of Cl to F in both samples will obey the law of multiple proportions.
C. The compound in Sample I is not the same as in Sample II. Therefore, the mass ratio of Cl to F in both samples will obey the law of definite proportions.
D. The compound in Sample I is not the same as in Sample II. Therefore, the mass ratio of Cl to F in both samples will obey the law of multiple proportions.

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

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 $y\ge 0$

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

The graph of f is shown. Evaluate each integral by interpreting it in terms of areas. integral.${\int }_0^{2}f(x)dx$

Find a Cartesian equation for the curve and identify it.
$r=7\cos (\theta )$

Consider an object moving along the parametrized curve with equations:
$x(t)=e^t+e^{-t}$
$y(t)=e^{-t}$
where t is in the time interval [0,4] seconds.
Fill in the blanks below.
The maximum speed of the object on the time interval is ____ at time ____.
The minimum speed of the object on the time interval is ____ at time ____.

Calculate the double integral $\frac{6x}{1+xy}dA,R=[0,6]\times [0,1]$

Calculate the iterated integral.
${\int }_0^3{\int }_0^{1}4xy(\sqrt{x^2+y^2})dydx$