Use the value of the first integral i to evaluate the two given integrals. i = integral^3_0 (x^3 - 4x)dx = 9/4 a. integral^3_0 (8x - 2x^3)dx b. integral^0_3 (4x - x^3)dx c. integral^3_0 (8x - 2x^3)dx =

A) -9/2

B) 9/4

C) -9/2, same as A)

Step-by-step explanation:

We are given that . We use the properties of integrals to write the new integrals in terms of I.

A) . We have used that ∫cf dx=c∫f dx.

B) . Here we used that reversing the limits of integration changes the sign of the integral.

C) It's the same integral in A)

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12

80π/3 or about 83.775

step-by-step explanation:

since a cone is 1/3 of a cylinder, we need to find the base's area. since area = πr^2, we get 16π. then, multiplying this by 5, we get 80π. dividing by 3, since the cone's volume is 1/3 of that of a cylinder's, we get 80π/3.

hope this !