Math

Here are the conversions from centimeters (cm) and meters (m) to millimeters (mm):

1. A) 1 cm: There are 10 mm in 1 cm.
2. B) 4 cm: There are 40 mm in 4 cm.
3. C) 0.5 cm: There are 5 mm in 0.5 cm.
4. D) 6.7 cm: There are 67 mm in 6.7 cm.
5. E) 1 m: There are 1000 mm in 1 m.

Conversion Factor: 1 cm = 10 mm, 1 m = 1000 mm

The volume of a cube is calculated by cubing the length of one of its sides. In other words, Volume = side * side * side.

Given a cube with sides of 3cm, the volume is:

Volume = 3cm * 3cm * 3cm = 27cm³

Therefore, the volume of the cube is 27 cubic centimeters.

To determine the quadrant in which an angle of -1560 degrees lies, we first need to find its coterminal angle within the range of 0 to 360 degrees.

We can do this by adding multiples of 360 to -1560 until we get an angle in the desired range:

-1560 + 360 = -1200

-1200 + 360 = -840

-840 + 360 = -480

-480 + 360 = -120

-120 + 360 = 240

So, -1560 degrees is coterminal with 240 degrees.

Now we determine the quadrant:

• Quadrant I: 0° to 90°
• Quadrant II: 90° to 180°
• Quadrant III: 180° to 270°
• Quadrant IV: 270° to 360°

Since 240° is between 180° and 270°, the angle lies in Quadrant III.

To find the number of integers between 30 and 50, inclusive, we can subtract the lower bound from the upper bound and add 1.

So, the calculation is:

50 - 30 + 1 = 21

Therefore, there are 21 numbers from 30 to 50, inclusive.

The integral of f(x) * g(x) dx does not have a general, simple formula like the integral of a sum or a constant multiple. Instead, it is usually solved using a technique called integration by parts.

Integration by parts comes from the product rule for differentiation. The formula is:

Where:

• u = f(x) or g(x) (a function to be differentiated)
• dv = the remaining part of the integrand, including dx (a function to be integrated)
• du = derivative of u
• v = integral of dv

Steps to apply integration by parts:

1. Choose u and dv from the integrand f(x)g(x) dx. A helpful guideline is the acronym LIATE:
   • Logarithmic functions
   • Inverse trigonometric functions
   • Algebraic functions
   • Trigonometric functions
   • Exponential functions
   Choose u based on what comes first in LIATE. The rest becomes dv.
2. Calculate du (the derivative of u) and v (the integral of dv).
3. Apply the integration by parts formula: ∫ u dv = uv - ∫ v du
4. Evaluate the new integral ∫ v du. If it's simpler than the original, the choice of u and dv was good. If not, you might need to try a different choice for u and dv or apply integration by parts again.

Example:

Evaluate ∫ x cos(x) dx

1. Let u = x (algebraic) and dv = cos(x) dx (trigonometric).
2. Then du = dx and v = ∫ cos(x) dx = sin(x).
3. Apply the formula: ∫ x cos(x) dx = x sin(x) - ∫ sin(x) dx
4. Evaluate the new integral: ∫ sin(x) dx = -cos(x)
5. Therefore, ∫ x cos(x) dx = x sin(x) - (-cos(x)) + C = x sin(x) + cos(x) + C, where C is the constant of integration.

In summary, there is no direct general formula for the integral of f(x) * g(x) dx. You typically need to use integration by parts.