Using the general slicing method to find the volume of a semi-circle whose cross sections are squares.

In finding the volume of a solid, described below, I was close in finding the equation, but neglected a coefficient. Please see the question below. Use the general slicing method to find the volume of the following solid. The solid with a semicircular base of radius 8 whose cross sections, perpendicular to the base and … Read more

Calculating the volume with double integral

Hello i am trying to calculate the volume for a double integral but i am having problem with define the integral because it is not given in a pure form. I have z=xy, x+y+z=1 z=0 my approach is to set the function for a integral to be ∫Dxy and to find the limitsfor dy i … Read more

Volume generated by rotating around y-axis, curve y=x3y=x^3 and the lines y=0y=0 and x=2x=2

Find the volume of the solid generated by revolving about the y-axis the region bounded by the curve y=x3 and the lines y=0 and x=2 I first found what x=2 would be in terms of y. y=(2)3=8 And in terms of y, the original equation becomes: y=x3x=y13 So, V=∫80(y13)2πdy=π∫80y23dy=π[y53(35)]80=(853)(35)π=965π Therefore the answer is 965π units … Read more

Calculate the volume between $z=\sqrt{x^2+y^2}$ and $z=x^2+y^2$.

Calculate the volume between $z=\sqrt{x^2+y^2}$ and $z=x^2+y^2$. Attempt We project on the $xy$ plane the intersection between $z=\sqrt{x^2+y^2}$ and $z=x^2+y^2$, which is the circle $x^2+y^2=1, z=1$. We can conclude that the region between $z=\sqrt{x^2+y^2}$ and $z=x^2+y^2$ can be described by $$-1\leq x\leq 1, -\sqrt{1-x^2}\leq y \leq \sqrt{1-x^2}, x^2+y^2\leq z \leq \sqrt{x^2+y^2}$$ The volume is given … Read more

Volume of regular tetrahedron in a cube

Given a cube with side length $a$, a regular tetrahedron is constructed such that two vertices of the tetrahedron lie on the cube’s body diagonal and the other two vertices lie on the diagonal of one of the faces of the cube. Determine the volume of the tetrahedron. All I can do is to visualise … Read more

Finding the volume of the tetrahedron.

Find the volume of the tetrahedron with the vertices $P(1,1,1)$, $Q(1, 2, 3)$, $R(3, 1, 2)$, and $S(2, 3, 1)$. Answer Volume of a tetrahedron is $$\dfrac13 \times \text{Base area} \times \text{height}$$ If the vertices are $\vec{v_1},\vec{v_2},\vec{v_3},\vec{v_4}$, then the volume is given by $$\left \vert \dfrac{(\vec{v}_2 – \vec{v}_1) \cdot \left((\vec{v}_3 – \vec{v}_1) \times (\vec{v}_4 – … Read more

Negative Volume using A⋅(B×C)\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})

So, my textbook explains how to find the volume of a paralelpiped using A⋅(B×C). Makes sense, basically. But, when I go to do problems some combinations produce negative volumes. Example: P(−2,1,0), Q(2,3,2), R(1,3,−1), S(3,6,1) and I compute the following vectors: PQ=<4,2,2> PR=<3,3−1> PS=<5,5,1>. However, this creates a little problem: PR⋅(PQ×PS)=−16 whereas PQ⋅(PR×PS)=16. I know that the volume is 16, but … Read more

Volume of Intersection of Surfaces

I hope you can help me with this: What’s the volume which is enclosed by the equation (x2+y2+z2)2=z(x2+y2)? Whenever I try to calculate the intersection of those surfaces I get lost because I arrive to a 4-degree equation. Thanks! Answer I think the simplest method is to use spherical coordinates x=rcosφsinθ,y=rsinφsinθ,z=rcosθ. Then the equation (x2+y2+z2)2=z(x2+y2) … Read more