8. Convert the following integrals from cartesian to the indicated coordinate sy

Question

8. Convert the following integrals from cartesian to the indicated coordinate systems: (a) 8 9x2 (x2 + y2) dy dx into polar coordinates. 01 8 9y2 9y2z2 (x + z) dx dz dy into cylindrical and spherical coordinates. (b)

9. Write down an equation for the following objects: 8 1 9y2z2 (a) a plane normal to 3i + 5k containing the point (1,2,3). (b) a plane with x-slope = 1, y-slope = 2 and z-intercept = 3. (c) a line parallel to 3i + 5k containing the point (1,2,3). (d) a circle of radius 3, centered at (1,2,3) on a plane parallel to the xz-plane, traversed clockwise when viewed from positive y-axis. (Note the orientation of the axis in 3-dimensions!)

7. set up iterated integals for the following double and triple integrals in cartesians: (a) R f(r,y) dA ents the total mass of a planar substrate R, with density function f( av, R is region bounded between ar 4 and the y-axis. (b) Jw dv represents the volume of solid w, where w is region bounded below by 2 z2 +F, and above 8. Convert the following integrals from cartesian to the indicated coordinate systems: (r2 y2) dy dr into polar coordinates. (ar 2) dar dz dy into cylindrical and spherical coordinates. (b) 9. write down an equation for the following objects: (a) a plane normal to 3i+ 5K containing the point (1,2,3). (b) a plane with T-slope 1, y-slope 2 and 2-intercept 3. (c) a line parallel to 3i+ 5k containing the point (1,2,3). (d) a circle of radius 3, centered at (1,2,3) on a plane parallel to the zz-plane, traversed clockwise when viewed from positive v-axis. (Note the orientation of the axis in 3-dimensions!)

Explanation / Answer

8a)

The general equation of the plane which is perpendicular to a given vector 'n is :

n * (X - x, Y - y, Z - z) = 0, where (x, y, z) is the given point.

So, in our case, x = 1, y = 2, z = 3

So, equation of plane is :

(3, 5, 0) * (X - 1, Y - 2, Z - 3) = 0,

or

3(X - 1) + 5(Y - 2) - 0(Z - 3) = 0

or

3X + 5Y - 13 = 0 ANS

8b)

We know that general equation of a plane is given by:

ax + by + cz = 0, where a, b, c are direction ratios or slopes along x, y, z axes respectively.

in our question, we have x-slope = 1, y-slope = 2 and z-intercept = 3, so our equation becomes :

1*0 - 2*0 + c*3 = 0, implies c = 0 [at z = 3 point, x = y = 0]

So, our equation is : 1.x -2.y + 0.z = 0

or x - 2y = 0.

8c)

In general, we know that the equation of the line passing through the point (x1,y1,z1) & parallel to the vector (ai+bj+ck) is given by

(x - x1)/a = (y - y1)/b = (z - z1)/c

(x - 1)/3 = (z - 3)/5 ANS

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