Question 1

Find the area enclosed by the given curves.
-Find the area between the curves y = ln x and y = ln 3x from x = 1 to x = 6.

Accepted Answer

The area between the curves y = ln x and y = ln 3x from x = 1 to x = 6 can be found by integrating the difference between the two functions over the given interval. The integral of (ln 3x - ln x) from 1 to 6 equals the integral of ln(3x/x) = ln 3 from 1 to 6, which simplifies to ln 3 times (6 - 1) = 5ln 3 = ln(3^5) = ln 243.

Question 2

Answer each question appropriately.
-$$\text { Suppose that } \int _ { 0 } ^ { 3 } f ( x ) d x = 11 \text {. Find } \int _ { - 3 } ^ { 0 } f ( x ) d x \text {, if } f \text { is even. }$$

Accepted Answer

If the function is even, then $\int_{-3}^{0} f(x) dx = \int_{0}^{3}f(-x) dx$. Since we know $\int_{0}^{3}f(x)dx = 11$, then $\int_{0}^{3}f(-x)dx = 11$. We can make the substitution $u = -x$, so that $dx = -du$ and the limits of integration change to $-3$ and $0$. Thus, $\int_{-3}^{0}f(x)dx = \int_{3}^{0}f(u)(-du) = \int_{0}^{3}f(u)du = \int_{0}^{3}f(x)dx = 11$.

Question 3

Answer each question appropriately.
-$$\text { Suppose that } \int _ { 0 } ^ { 5 } \mathrm { f } ( \mathrm { x } ) \mathrm { dx } = 10 \text {. Find } \int _ { - 5 } ^ { 0 } \mathrm { f } ( \mathrm { x } ) \mathrm { dx } \text {, if } \mathrm { f } \text { is odd. }$$

Accepted Answer

Since $\mathrm{f(x)}$ is odd, we know that   $\int_{-a}^{0} \mathrm{f(x)} \mathrm{dx} =-\int_{0}^{a} \mathrm{f(x)} \mathrm{dx}$. Thus, $\int_{-5}^{0}\mathrm{f(x)}\mathrm{dx}=-\int_{0}^{5}\mathrm{f(x)}\mathrm{dx}=-10$.

Question 4

Answer each question appropriately.
-Suppose the velocity of a body moving along the $s$-axis is $\frac { \mathrm { ds } } { \mathrm { dt } } = 9.8 \mathrm { t } - 10$.
Is it necessary to know the initial position of the body to find the body's displacement over some time interval? Justify your answer.

Accepted Answer

The initial position is necessary to find the curve s=f(t) but not necessary to find the displacement. When finding the displacement, the integration constant is subtracted out.

Question 5

Find the area enclosed by the given curves.
-Find the area of the region between the curve $y = 4 x / \left( 1 + x ^ { 2 } \right)$ and the interval $- 3 \leq x \leq 3$ of the $x$-axis.

Accepted Answer

To find the area enclosed by the curve, we need to integrate the function from $x=-3$ to $x=3$.

$$A=\int_{-3}^{3} \frac{4x}{1+x^2} dx$$

We can solve this integral by substituting $u$ = 1+$x^2$. Then the integral becomes:

\begin{align*}
\int \frac{4x}{1+x^2} dx &=\ln|1+x^2|+C\
&=\ln(1+3^2)-\ln(1+(-3)^2) \
&= \ln(10)
\end{align*}

Therefore, the area enclosed by the curve is $\ln(10)$, which is option C.

Question 6

Answer each question appropriately.
-If differentiable functions $y = F ( x )$ and $y = G ( x )$ both solve the initial value problem $\frac { d y } { d x } = f ( x ) , \quad y \left( x _ { 0 } \right) = y _ { 0 }$, on an interval I, must $F ( x ) = G ( x )$ for every $x$ in I? Justify the answer.

Accepted Answer

Given an initial value problem with a specific initial condition, the solution to the differential equation is unique. This is because the initial condition allows for the determination of the integration constant, ensuring that any two functions $F(x)$ and $G(x)$ that satisfy both the differential equation and the initial condition must be identical on the interval $I$.

Question 7

Answer each question appropriately.
-Which of the following integrals, if any, calculates the area of the shaded region?

Accepted Answer

The answer of Answer each question appropriately.
-Which of the following...

Question 8

Find the area enclosed by the given curves.
-Find the area of the region in the first quadrant bounded on the left by the $y$-axis, below by the line $y = \frac { 1 } { 3 } x$, above left by $y = x + 4$, and above right by $y = - x ^ { 2 } + 10$.

Accepted Answer

The answer of Find the area enclosed by the given...

Question 9

Find the area enclosed by the given curves.
-Find the area of the "triangular" region in the first quadrant that is bounded above by the curve $y = e ^ { 2 x }$, below by the curve $y = e ^ { x }$, and on the right by the line $x = \ln 5$.

Accepted Answer

The answer of Find the area enclosed by the given...

Question 10

Answer each question appropriately.
-Suppose the area of the region between the graph of a positive continuous function f and the x-axis from x = a to x = b is 13 square units. Find the area between the curves y = f(x) and y = -3f(x).

Accepted Answer

The area between the curves y = f(x) and y = -3f(x) is given by:

∫[a,b] (-3f(x) - f(x)) dx

= ∫[a,b] (-2f(x)) dx

= -2∫[a,b] f(x) dx

The negative sign indicates that the area is below the x-axis. Since the given function f(x) is positive, the area must be negative. Let the area be -A.

So, we have:

-A = 13

A = -13

The area between the curves is -2A = 26 square units, which is positive because of the negative sign outside. Therefore, the answer is D.

Question 11

Find the area enclosed by the given curves.
-Find the area of the region between the curve $y = 5 ^ { 2 - x }$ and the interval $0 \leq x \leq 2$ on the $x$-axis.

Accepted Answer

The area under the curve $y = 5^{2-x}$ from $x = 0$ to $x = 2$ can be found by integrating the function with respect to $x$ over the given interval. The integral of $5^{2-x}$ with respect to $x$ is $-\frac{5^{2-x}}{\ln 5}$, and evaluating this from $0$ to $2$ gives $\left[-\frac{5^{2-2}}{\ln 5} - \left(-\frac{5^{2-0}}{\ln 5}\right)\right] = \frac{24}{\ln 5}$.

Question 12

Answer each question appropriately.
-Suppose the area of the region between the graph of a positive continuous function f and the x-axis from x = a to x = b is 11 square units. Find the area between the curves y = f(x) and y = 5f(x).

Accepted Answer

The answer of Answer each question appropriately.
-Suppose the area of...

Quiz 6: Integrals

Find the area enclosed by the given curves. -Find the area between the curves y = ln x and y = ln 3x from x = 1 to x = 6.

Answer each question appropriately. - $\text { Suppose that } \int _ { 0 } ^ { 3 } f ( x ) d x = 11 \text {. Find } \int _ { - 3 } ^ { 0 } f ( x ) d x \text {, if } f \text { is even. }$

Answer each question appropriately. - $\text { Suppose that } \int _ { 0 } ^ { 5 } \mathrm { f } ( \mathrm { x } ) \mathrm { dx } = 10 \text {. Find } \int _ { - 5 } ^ { 0 } \mathrm { f } ( \mathrm { x } ) \mathrm { dx } \text {, if } \mathrm { f } \text { is odd. }$

Find the area enclosed by the given curves. -Find the area of the region between the curve $y = 4 x / \left( 1 + x ^ { 2 } \right)$ and the interval $- 3 \leq x \leq 3$ of the $x$ -axis.

Answer each question appropriately. -If differentiable functions $y = F ( x )$ and $y = G ( x )$ both solve the initial value problem $\frac { d y } { d x } = f ( x ) , \quad y \left( x _ { 0 } \right) = y _ { 0 }$ , on an interval I, must $F ( x ) = G ( x )$ for every $x$ in I? Justify the answer.

Find the area enclosed by the given curves. -Find the area of the region in the first quadrant bounded on the left by the $y$ -axis, below by the line $y = \frac { 1 } { 3 } x$ , above left by $y = x + 4$ , and above right by $y = - x ^ { 2 } + 10$ .

Find the area enclosed by the given curves. -Find the area of the "triangular" region in the first quadrant that is bounded above by the curve $y = e ^ { 2 x }$ , below by the curve $y = e ^ { x }$ , and on the right by the line $x = \ln 5$ .

Answer each question appropriately. -Suppose the area of the region between the graph of a positive continuous function f and the x-axis from x = a to x = b is 13 square units. Find the area between the curves y = f(x) and y = -3f(x) .

Find the area enclosed by the given curves. -Find the area of the region between the curve $y = 5 ^ { 2 - x }$ and the interval $0 \leq x \leq 2$ on the $x$ -axis.

Answer each question appropriately. -Suppose the area of the region between the graph of a positive continuous function f and the x-axis from x = a to x = b is 11 square units. Find the area between the curves y = f(x) and y = 5f(x) .

Quiz 6: Integrals

Find the area enclosed by the given curves. -Find the area between the curves y = ln x and y = ln 3x from x = 1 to x = 6.

Find the area enclosed by the given curves. -Find the area of the region between the curve y=4x/(1+x2) y = 4 x / \left( 1 + x ^ { 2 } \right) y=4x/(1+x2) and the interval −3≤x≤3- 3 \leq x \leq 3−3≤x≤3 of the xxx -axis.

Answer each question appropriately. -Which of the following integrals, if any, calculates the area of the shaded region?

Find the area enclosed by the given curves. -Find the area of the region in the first quadrant bounded on the left by the yyy -axis, below by the line y=13xy = \frac { 1 } { 3 } xy=31​x , above left by y=x+4y = x + 4y=x+4 , and above right by y=−x2+10y = - x ^ { 2 } + 10y=−x2+10 .

Find the area enclosed by the given curves. -Find the area of the "triangular" region in the first quadrant that is bounded above by the curve y=e2xy = e ^ { 2 x }y=e2x , below by the curve y=exy = e ^ { x }y=ex , and on the right by the line x=ln⁡5x = \ln 5x=ln5 .