Solve Integral Calculus Problems with Ease

College
Calculus and Analysis
Integral Calculus

How do you find the equation of the tangent line to the curve given by parametric equations: $x = 1 + (\frac{1}{t^{2}})$ , $y = 1 - (\frac{3}{t})$ at the point when t=2?

Andy Erickson 2022-05-21

Parametric Function for Coloring
I'm trying to write a parametric straightline function that changes its values between 0.529 and 0.933.
Usually what I would do is:
$r = 0.529 + (0.933 - 0.529) * v$ where parameter $v = [0, 1]$
This ensure that my r value will either be 0.529 or 0.933
However, due to some limitations my parameter v has to be changed to $v = [- 2, 1]$
Is there anyway that I can change the straightline function such that it will still return r as either 0.529 or 0.933?

reryfaikear 2022-05-20

I have the folllwoing integral
$\int_{0}^{π / 4} \int_{0}^{2 π} \sin^{2} (ρ) \sin (θ) - \cos (ρ) \sin^{2} (ρ) \sin (θ) d θ d ρ$
and I would like to know why exaclty one can rewrite it as follows
$(\int_{0}^{π / 4} \sin^{2} (ρ) - \cos (ρ) \sin^{2} (ρ) d ρ) \int_{0}^{2 π} \sin θ d θ .$

hughy46u 2022-05-20

What is the polar form of (19, 55)?

Anahi Jensen 2022-05-20

Getting a differential equation from a parametric function family
$x = c e^{- t} - 2 t + 2 y = c (1 + t) e^{- t} - t^{2} + 2$

Isaiah Farrell 2022-05-20

For $f (t) = (\sin 2 t, \cos^{2} t)$ what is the distance between $f (\frac{π}{4})$ and $f (π)$

skottyrottenmf 2022-05-19

Compute
$\iint_{D} \frac{x^{2}}{\sqrt{1 + x^{2} + y^{2}}} d x d y$
where $D = {(x, y) | x^{2} + y^{2} \leq 1, x \geq 0}$

Nylah Burnett 2022-05-19

What is the polar form of $(11, - 8)$ ?

madridomot 2022-05-19

What is the distance between the following polar coordinates?:
$(10, \frac{17 π}{12}), (4, \frac{15 π}{8})$

agdv9m 2022-05-19

What is the distance between the following polar coordinates?:
$(- 3, \frac{7 π}{4}), (5, \frac{5 π}{8})$

Laila Andrews 2022-05-17

How do you write the corresponding rectangular equation by eliminating the parameter given $x = 3 t - 1, y = 2 t + 1$ ?

lifretatox8n 2022-05-17

How do you write a vector equation and a parametric equation for each line: the line through $A (1, - 3, 1)$ and parallel to vector $u = (2, - 2, 1)$ ?

Elliptic integrals The length of the ellipse
x=acost,y=bsint,0≤t≤2π $x = a \cos t, y = b \sin t, 0 \leq t \leq 2 π$
turns out to be
=4a∫π/201−e2cos2t√dt $= 4 a \int_{0}^{π / 2} \sqrt{1 - e^{2} \cos^{2} t} d t$
where e $e$ is the ellipse's eccentricity. The integral in this formula, called an elliptic integral, is non elementary except when e=0 $e = 0$ or 1 a. Use the Trapezoidal Rule with n=10 $n = 10$ to estimate the length of the ellipse when a=1 $a = 1$ and e=1/2 $e = 1 / 2$ . b. Use the fact that the absolute value of the second derivative of f(t)=1−e2cos2t√ $f (t) = \sqrt{1 - e^{2} \cos^{2} t}$ is less than 1 to find an upper bound for the error in the estimate you obtained in part (a).

sg101cp6vv 2022-05-15

For $f (t) = (\frac{\ln t}{e^{t}}, \frac{e^{t}}{t}) what is the distance between f (1)$ and $f (2)$

Alisa Durham 2022-05-15

Antiderivative of a trigonometric integral
$\int \frac{\sin^{3} (x)}{(\cos^{4} (x) + 3 \cos^{2} (x) + 1) \cdot \arctan (\sec (x) + \cos (x))}$
I am unable to manipulate this integral. The actual integral in question is definite with limits from 0 to $\frac{π}{2}$ , but I feel it can't be calculated without knowing the antiderivative.
My initial try was to take $\sec (x) + \cos (x)$ as t, but It didn't solve.

indimiamimactjcf 2022-05-15

Compute the volume enclosed by the surface
${(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}})}^{2} = \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} .$

indimiamimactjcf 2022-05-15

What is the polar form of $(5, 114)$ ?

othereyeshmt4l 2022-05-15

What is the polar form of (3, 27)?

Oberhangaps5z 2022-05-15

How do you write the parametric equations represent the ellipse given by $\frac{x^{2}}{9} + \frac{y^{2}}{81} = 1$

Kendall Oneill 2022-05-15

How to change the order of integration:
$\int_{- 1}^{1} d x \int_{1 - x^{2}}^{2 - x^{2}} f (x, y) d y$

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Integral Calculus

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