1. Find the area of the region bounded by the curves and lines y=e^x sin e^x, x=0, y=0, and the curve's first positive intersection with the x-axis.

2. The area under the curve of y=1/x from x=a to x=5 is approximately 0.916 where 1<=a<5. Using your calculator, find a.

3. Let R be the region bounded by y=6e^(-.2x) and y=√x and the lines x=1 and x=4. Find the volume when R is rotated about the x-axis.

4. Find the value(s) of b if the vertical line x=b divides the region between y=16-2x and the x and y-axis into 2 equal areas.

Please show work.

1. y=e^x sin e^x

Naturally, the first place where y=0 is at e^x=π, or x = ln π.

So, the area is just
∫[0,π] e^x sin e^x dx
That's just sin u du, so the value is 1 + cos(1)

2. just solve ln5 - ln a = .916

3. using shells,
v = ∫[1,4] 2πrh dx
where r = 1+x and h = y = 6e^(-.2x)-√x
Pretty easy integration

4. the region is just a triangle, so you want the area of the trapezoid on 0<x<b to equal the triangle on b<x<8

b(16+y(b))/2 = 1/2 (8-b)*y(b)

evaluate y at x=b, plug it in, and solve for b

1. To find the area of the region bounded by the curves and lines, we need to find the points of intersection first.

Setting y = 0 in the equation y = e^x sin(e^x), we have:
e^x sin(e^x) = 0

This equation holds true when either e^x = 0 or sin(e^x) = 0.

e^x = 0 has no solutions since e^x is always positive.

Next, we solve for sin(e^x) = 0:
e^x = nπ, where n is an integer.

Taking the natural logarithm (ln) of both sides:
x = ln(nπ)

Since we are looking for the first positive intersection, we set n = 1:
x = ln(π)

Now we can find the area of the region by integrating y = e^x sin(e^x) from x = 0 to x = ln(π):
Area = ∫[0, ln(π)] (e^x sin(e^x)) dx

2. To find the value of a, we need to calculate the definite integral.

The integral ∫[a, 5] (1/x) dx gives the area under the curve y = 1/x from x = a to x = 5.

Given that the area is approximately 0.916, we can set up the equation:
0.916 = ∫[a, 5] (1/x) dx

To solve this equation, we can use numerical methods or a calculator that can perform numerical integration. By inputting the equation into the calculator, we can find the value of a.

3. To find the volume when region R is rotated about the x-axis, we'll use the method of cylindrical shells.

The volume of the solid generated is given by the integral:
Volume = ∫[1, 4] [(2πx)(6e^(-0.2x) - √x)] dx

We subtract √x since the region is bounded by two curves.

Evaluate this integral to get the volume.

4. To find the value(s) of b, we need to determine the area of the region between y = 16 - 2x and the x and y-axis, and then find the value of b that divides this area into two equal parts.

First, we need to find the points of intersection between the curve y = 16 - 2x and the x-axis.

Setting y = 0 in the equation 16 - 2x = 0:
x = 8

So the region spans from x = 0 to x = 8.

To find the area of the region, we integrate the equation y = 16 - 2x from x = 0 to x = 8:
Area = ∫[0, 8] (16 - 2x) dx

Next, we can find the value(s) of b such that the area between y = 16 - 2x, the x-axis, and the vertical line x = b is equal to half of the total area.

∫[0, b] (16 - 2x) dx = (1/2) * Area

Evaluate this integral and solve for b to find the value(s).

1. To find the area of the region bounded by the curves and lines y = e^x sin(e^x), x = 0, y = 0, and the curve's first positive intersection with the x-axis, you can use the definite integral.

First, find the x-coordinate of the curve's first positive intersection with the x-axis. Set y = 0 and solve the equation e^x sin(e^x) = 0 for x. This can be done by finding the values of x for which sin(e^x) = 0. Since sin(theta) = 0 when theta is a multiple of pi, we have e^x = n*pi, where n is an integer. Solve for x to find the first positive x-intersection.

Let's say this x-coordinate is a positive value x = c.

The area of the region can be found by integrating the curve between x = 0 and x = c. Use the following integral to calculate the area:

Area = ∫[0, c] (e^x sin(e^x)) dx

Use any numerical method of integration, such as numerical approximation or calculus software, to evaluate the integral and find the area.

2. The area under the curve y = 1/x from x = a to x = 5 is approximately 0.916 where 1 ≤ a < 5. To find the value of a, you can use numerical methods or calculus techniques.

One way to find the value of a is to set up an equation using the definite integral of the function and solve for a.

The area under the curve is given by the integral:

Area = ∫[a, 5] (1/x) dx

Evaluate this integral to obtain the expression for the area in terms of a:

Area = ln(5) - ln(a)

Then, set this expression equal to the given approximate area and solve for a:

ln(5) - ln(a) = 0.916

Solve this equation to find the value of a.

Alternatively, you can graph the function y = 1/x and visually estimate the value of a where the area under the curve is approximately 0.916 using a calculator or graphing software.

3. To find the volume when the region R, bounded by y = 6e^(-0.2x), y = √x, x = 1, and x = 4, is rotated about the x-axis, you can use the method of cylindrical shells or the disk method.

Using the disk method, the volume can be calculated by integrating the cross-sectional areas of infinitesimally thin disks perpendicular to the x-axis.

The cross-sectional area of each disk can be defined as the difference between the areas of the outer curve and the inner curve. In this case, the outer curve is y = 6e^(-0.2x) and the inner curve is y = √x.

The volume can be calculated using the integral:

Volume = ∫[1, 4] (π[r_outer(x)]^2 - π[r_inner(x)]^2) dx

where r_outer(x) is the radius of the outer curve and r_inner(x) is the radius of the inner curve.

Substituting the equations for y = 6e^(-0.2x) and y = √x into the integral, the expression for volume can be obtained.

Evaluate this integral to find the volume of the solid.

4. To find the value(s) of b if the vertical line x = b divides the region between y = 16 - 2x and the x and y-axis into two equal areas, you can use the concept of finding the centroid or balancing point of the region.

The area of the region between the curves y = 16 - 2x and the x and y-axis can be found by integrating the difference between the functions from their points of intersection.

Set y = 16 - 2x equal to 0 to find the x-coordinate of the intersection point. Solve for x to find the value of b.

Then, evaluate the integral:

Area = ∫[0, b] (16 - 2x) dx

Integrate the expression to find the value of the area.

Next, divide the area by 2 to find the equal areas on each side of the vertical line x = b.

Set up the equation:

Area/2 = ∫[0, b] (16 - 2x) dx

Integrate the expression and solve for b to find the value(s) that divide the region into two equal areas.