Understanding Engineering Mathematics

If we rotate a thin strip of area,dxwide, about the axis, then a thin disc is formed, whose radius isr=y, and thickness ish=dx. ...

10.3.2 Tangent and normal to a curve ➤➤ 291 293 ➤ Find the equations of the tangents and the normals to the following curves at ...

B. Calculate the total signed area between the curves and thex-axis and the limits given by: (a) geometry, (b) integration. (i) ...

(i) An oil spill from a ruptured tanker in calm seas spreads out in a circular pattern with the radius increasing at a constant ...

mass of a lamina with surface densityρ(x,y)enclosed within the curvey=f(x),the x-axis and the linesx=a,x=b, are given by x ̄= ∫b ...

(ii) Calculate the moment of inertia about the coordinate axes of (i) y=x^2 , y=0,x= 4 (ii) 2 y=x^3 ,x= 0 y= 4 (iii) xy=4, x=1,y ...

(iii) Point of inflection at (0, 0) (iv) Min at (5, 2), max at (−5,−2) (v) Min at (3, 23), max at (2, 24), point of inflection a ...

10.3.6 Volume of a solid of revolution (i) π 30 (ii) 3 π 10 (iii) π^2 2 ...

11 Vectors Vectors are the mathematical tools needed when we deal with engineering systems in more than one dimension. Vector qu ...

properties of vectors the scalar product of two vectors the vector product of two vectors vector functions differentiation of v ...

and ‘vector’ are also used in matrix theory, and indeed matrices can be used to represent vectors – a 3×1 column matrix can be u ...

Thelengthormagnitudeof a vector,a, is denoted|a|or just a. The direction ofais represented by a vector in the same direction, bu ...

11.3 Addition and subtraction of vectors Geometrically, vectors areaddedby the use of thetriangle lawin whichaandbform two sides ...

a a + b + c + d + e + f + g b c e f g d Figure 11.4Polygon addition of vectors. A particular example of this isscalar multiplica ...

Answers (i)e=−(a+b+c+d) (ii) −→ CE=−(e+a+b) (iii) −→ CE=c+d (iv) −→ EC= −→ EA− −→ CA 2.b−ais of length √ 3 parallel to they-ax ...

Answer y x z (0,0,0) (0,1,0) (0,0,1) (2,−1,1) (−1,0,0) (1,0,0) (1,0,−1) 11.5 Distance in Cartesian coordinates Look again at Fig ...

Exercise on 11.5 Calculate the distances of the points (i) (1, 0, 2), (ii) (−2, 1,−3), (iii) (−1,−1,−4) from the origin. Also ca ...

from which, since r^2 =x^2 +y^2 +z^2 , l^2 +m^2 +n^2 = 1 Note that the direction cosines of a line not passing through the origi ...

obtained as (cos 45°, 0 ,cos 45°)= ( 1 √ 2 , 0 , 1 √ 2 ) This is of course what we would obtain from the coordinates (1, 0, 1) t ...

A′ B′(l′,m′,n′) B(l,m,n) q A 1 O 1 Figure 11.9Angle between two lines. This follows by applying the cosine rule to the triangleO ...