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snow_tiger [21]
3 years ago
15

Find the volume of the solid formed by rotating the region bounded by the given curves about the indicated axis. y = x3/4, x = 0

, y = 1 (in the first quadrant); about the y-axis
Mathematics
1 answer:
vovangra [49]3 years ago
3 0

Answer:

7.92 cubic units

Step-by-step explanation:

The volume formed is V = ∫πx²dy

Now, since y = x³/4, x = ∛(4y). Also if x = 0, y = 0³/4 = 0 and the curve intersects the line y = 1. So the limits of integration are y = 0 to y = 1

So, V = ∫₀¹πx²dy

= ∫₀¹π[ ∛(4y)]²dy

= π(∛4)²∫₀¹[ ∛(y]²dy

= π(∛4)²∫₀¹y^³/₂dy

= π(∛4)²[y^⁵/₂]₀¹

= π(∛4)²[1^⁵/₂ - 0^⁵/₂]

= π(∛4)²[1 - 0]

= π(∛4)²

= 7.92 cubic units

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Dima020 [189]

Split C into two component segments, C_1 and C_2, parameterized by

\mathbf r_1(t)=(1-t)(0,0)+t(6,1)=(6t,t)

\mathbf r_2(t)=(1-t)(6,1)+t(7,0)=(6+t,1-t)

respectively, with 0\le t\le1, where \mathbf r_i(t)=(x(t),y(t)).

We have

\mathrm d\mathbf r_1=(6,1)\,\mathrm dt

\mathrm d\mathbf r_2=(1,-1)\,\mathrm dt

where \mathrm d\mathbf r_i=\left(\dfrac{\mathrm dx}{\mathrm dt},\dfrac{\mathrm dy}{\mathrm dt}\right)\,\mathrm dt

so the line integral becomes

\displaystyle\int_C(x+6y)\,\mathrm dx+x^2\,\mathrm dy=\left\{\int_{C_1}+\int_{C_2}\right\}(x+6y,x^2)\cdot(\mathrm dx,\mathrm dy)

=\displaystyle\int_0^1(6t+6t,(6t)^2)\cdot(6,1)\,\mathrm dt+\int_0^1((6+t)+6(1-t),(6+t)^2)\cdot(1,-1)\,\mathrm dt

=\displaystyle\int_0^1(35t^2+55t-24)\,\mathrm dt=\frac{91}6

6 0
3 years ago
Please explain I need help with all these
sergey [27]
For all of these you're trying to solve for the variable. The variable is usually a number or symbol.

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Step one would be to add 44 to both sides of the equation.

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Step one would be to add -91 to both sides of the equation.

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13) X-225=671
Step one you be to add 225 to both sides of the equation. Although it may seem difficult because of the big numbers, it's the same technique.

14) 19=c-(-12)
Step one on this equation will be a little different. Because there are two negative signs (-), they turn into a plus sign. Remember, two negative signs make a positive sign.

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3 0
3 years ago
Please help asap i really need it.
pishuonlain [190]
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3 0
4 years ago
Question in pic, will mark brainliest.
bulgar [2K]

Answer:

See explanation

Step-by-step explanation:

Consider triangles ABC and DEC. In these triangles,

  • \overline{BC}\cong \overline {EC} - given
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So,

\triangle ABC\cong \triangle DEC by SAS postulate (two sides and angle between these sides of one triangle are congruent to two sides and angle between these sides of another triangle, so triangles are congruent).

Congruent triangles have congruent corresponding parts, hence,

\overline{BA}\cong \overline{ED}

3 0
3 years ago
PLEASE HELP
makvit [3.9K]

1st one is:SSS 2nd one is:*not visible* 3rd one is:SSA 4th one is: AAS, 5th one is:AAA or AAS 6th one is:CNBD 7th one is:SAS 8th one is:SSS

Step-by-step explanation:

Hope this helped :)

Tell me if anything is wrong please

7 0
3 years ago
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