Detailed Explanations for Calculus 2 Equations

How do you find the volume bounded by $y^{2} = x^{3}$ and $y = x^{2}$ revolved about the y-axis?

Ayanna Trujillo 2022-06-14

Most of us are aware of the classic Gaussian Integral
$\int_{0}^{\infty} e^{- x^{2}} d x = \frac{\sqrt{π}}{2}$

arridsd9 2022-06-14

I have this series:
$\sum_{n = 1}^{\infty} \frac{4}{(n + 1) (n + 2)}$

Tristian Velazquez 2022-06-14

How to prove this trigonometric integral?
$\int_{- π / 4}^{π / 4} {(\frac{\cos x - \sin x}{\cos x + \sin x})}^{\cos (2 t)} d x = \frac{π}{2 \sin (π \cos^{2} t)}$

manierato5h 2022-06-14

How would I simplify my definite integral properly?
Original equation
$\int_{1}^{3} \frac{1}{x^{2} - 2 x + 5} d x$

Finley Mckinney 2022-06-14

$f : R^{2} \to R$ is continuous and $f (t, x) < 0$ holds for $t x > 0$ and $f (t, x) > 0$ holds for $t x < 0$

I want to show, that $x (t) = 0$ is the only solution of $x^{'} = f (t, x)$

You can say, that there is an solution, because f is continuous. By the intermediate value theorem exists $ξ \in [- a, a]$ , such that $f (t, ξ) = 0$ . I only have to show, that $x (t) = 0$ is unique. How can I do that?

Eden Solomon 2022-06-13

Tips for integrating $\int \frac{d x}{1 + \cos (x)}$

Celia Lucas 2022-06-13

What is the equation for exponential decay?

gvaldytist 2022-06-13

How would I find the radius of convergence for $\sum_{n = 0}^{\infty} 2^{- n} z^{n^{2}}$ ? Im not sure how to deal with the $z^{n^{2}}$ term.

Dale Tate 2022-06-13

How mush work is done in lifting a 40 kilogram weight to a height of 1.5 meters?

Zion Wheeler 2022-06-13

Why is $\sum_{n = 1}^{\infty} \frac{1}{n} \sin (\frac{x^{n}}{\sqrt{n}})$ uniformly convergent?

Lydia Carey 2022-06-13

What is the arclength of $f (x) = x^{3} - e^{x}$ on $x \in [- 1, 0]$ ?

Jaqueline Kirby 2022-06-13

How do you find the average value of the function for $f (x) = \sqrt{2 x - 1}, 1 \leq x \leq 5$ ?

Leland Morrow 2022-06-13

Let $f$ be a continuously differentiable function on $[a, b]$ with $f (a) = f (b)$ and $f^{'} (a) = f^{'} (b)$ . Then, do there exist $x_{1}, x_{2} \in (a, b)$ such that $f^{'} (x_{1}) = f^{'} (x_{2})$ for $x_{1} \neq x_{2}$ ?

I think the answer is yes by intermediate value theorem, but am not getting the sufficient rigour to prove it. Any hints. Thanks beforehand.

excluderho 2022-06-13

Let $f : [a, b] \to R$ be continuous, $p, q > 0$ . We want to show that there is a $c \in [a, b]$ s.t. $p f (a) + q f (b) = (p + q) f (c)$ . $\frac{p}{p + q}, \frac{q}{p + q}$ are both in (0,1). Then $\frac{p}{p + q} f (a) + \frac{q}{p + q} f (b)$ is in $(0, f (a) + f (b))$ . If we want to apply the intermediate value theorem we better would need it to be in [ $f (u)$ , $f (w)$ ], where we know that $f (u)$ is the Minimum and $f (w)$ is the Maximum. But we have not any info about what the Min/Max would be?

Sunita Jain2022-06-12

Determine the largest interval in which the initial value problem (t − 3)y ′′ − ty′ + 6y = sin t has unique solution.
y(−3) = 2, y′ (−3) = 1

Brenden Tran 2022-06-12

What is the average value of a function y=50x+5 on the interval [3, 9]?

Mayra Berry 2022-06-12

What is the surface area of the solid created by revolving $f (x) = \sqrt{x^{3}}$ for $x \in [1, 2]$ around the x-axis?

Jeramiah Campos 2022-06-12

What is the volume of the solid produced by revolving $f (x) = \frac{1}{x}, x \in [1, 4]$ around the x-axis?

landdenaw 2022-06-12

What is the closed form of $\sum_{k = 0}^{n} (5^{k} + k^{3})$ ?

When you are dealing with any Calculus 2 homework, it is vital to have a look at the various questions and answers that will help you see whether you are correct in your approach to finding solutions. Even if you are dealing with analytical aspects of Calculus 2, it will be helpful as you are looking at provided equations and learn how the answers relate to original questions and problems specified.

Do not be afraid to take a look at the basic integration and related application if Calculus 2 does not sound clear or start with the Calculus 1 first.

Calculus 2

Calculus 2 Equations: Expert Solutions and More