Squaring

Solution:

Quadrature is the calculation of the area of a plane figure that is bounded by the x-axis, two lines parallel to the y-axis at the points x₁ = a, x₂ = b, and a line (straight or curved) with the equation y = f(x).

Solution:

a) Square

The square is bounded by the x-axis, the line y = a, and lines parallel to the y-axis at the points a = 0, b = a.

b) Rectangle

The rectangle is bounded by the x-axis, the line y = b, and lines parallel to the y-axis at the points a = 0, b = a.

Solution:

a) Isosceles triangle

The isosceles triangle has a base a and height v. Let S₁ be half of the area of this triangle, which is bounded by the x-axis, lines parallel to the y-axis at the points a = 0, b = v, and the line

b) Isosceles trapezoid

The isosceles trapezoid has an upper base a, a lower base b, and height v. Let S₁ be half of the area of this trapezoid, which is bounded by the x-axis, lines parallel to the y-axis at the points a = 0, b = v, and the line

Solution:

The intersections of the line y₁ = 5 and the parabola y₂ = x² + 1 are the limits of the integral.

Solution:

The intersections of the parabolas y₁ = 6 – 4x + x² and y₂ = –3 + 8x – 2x² are the limits of the integral.

Solution:

First, let us prove that the equations x = r·cos t and y = r·sin t are parametric equations of a circle.

Proof:

x² + y² = r²

r²cos²t + r²sin²t = r²

r²(cos²t + sin²t) = r²

r²·1 = r²

r² = r² – the statement holds

S₁ is one quarter of the circle, which is bounded by the x-axis, the y-axis (x = 0), and the circular arc y = r·sin t

Solution:

First, let us prove that the equations x = a·cos t and y = b·sin t are parametric equations of an ellipse.

Proof: