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3 years ago

Find the volume of the given solid. Under the surface z = 6xy and above the triangle with vertices (1, 1), (4, 1), and (1, 2)

Mathematics

1 answer:

Leya [2.2K]3 years ago

4 0

he volume of the solid under a surface

(

)

and above a region D is given by the formula

∫

(

)

Here

(

)

. The inequalities that define the region D can be found by making a sketch of the triangle that lies in the

−

plane. The bounding equations of the triangle are found using the point-slope formula as

and

−

Here is a sketch of the triangle:

Intersecting Region

The inequalities that describe D are given by the sketch as:

≤

and

≤

−

Therefore, volume is

∫

−

∫

[

]

−

∫

[

]

−

∫

[

−

]

∫

−

[

−

]

[

(

640

−

896

256

)

−

(

−

)

]

23.25

Volume is

23.25

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a(6) = 37.97

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2 years ago

If the parent function f(x) = x^2 is fatter translated 11 units to the left, then translated 5 units down, write the resulting f

nikklg [1K]

The quadratic function given by:

$f(x)=a(x-h)^2+k, \ \ \ a\neq 0$

is in vertex form. The graph of $f$ is a parabola whose axis is the vertical line $x=h$ and whose vertex is the point $(h, k)$ . So:

To translate the graph of a function to the right, left, upward or downward we have:

$For \ a \ positive \ real \ number \ c. \ \mathbf{Vertical \ and \ horizontal \ shifts} \\ in \ the \ graph \ of \ y=f(x) \ are \ represented \ as \ follows:\\ \\ \bullet \ Vertical \ shift \ c \ units \ \mathbf{upward}: \\ g(x)=f(x)+c \\ \\ \bullet \ Vertical \ shift \ c \ units \ \mathbf{downward}: \\ g(x)=f(x)-c \\ \\ \bullet \ Horizontal \ shift \ c \ units \ to \ the \ \mathbf{right}: \\ g(x)=f(x-c) \\ \\ \bullet \ Horizontal \ shift \ c \ units \ to \ the \ \mathbf{left}: \\ g(x)=f(x+c)$

By knowing this things, we can solve our problem as follows:

FIRST.

Translating 11 units to the left:

$g(x)=f(x+11) \\ \\ \therefore g(x)=(x+11)^2$

Then translating 5 units down:

$g(x)=f(x)-c \\ \\ \therefore g(x)=(x+11)^2-5$

Since the new function is fatter, the factor we need to multiply the term $(x+11)^2$ must be less than 1, to make the graph fatter. So, according to our options, there are two factors 1/2 and 2.