Integrals, Integral Theorems and Applications

Antiderivatives, definite integrals, area under and between curves, and recovering motion from velocity and acceleration.

11 questions, free, marked as you go with a worked explanation on every one.

What this covers

Evaluate the definite integral of 3x² with respect to x from x = 0 to x = 2.
Evaluate the definite integral of (4x³ - 6x) with respect to x from x = 1 to x = 3.
Find the area of the region bounded by the curve y = x², the x-axis, and the lines x = 0 and x = 3.
Find the area of the region enclosed between the curve y = 4 - x² and the line y = x + 2.
Why does an indefinite integral always include a constant of integration, + C?
A car's velocity after t seconds is v(t) = 6t + 2 metres per second. How far does the car travel between t = 0 and t = 4 seconds, in metres?
A curve has gradient dy/dx = 6x at every point and passes through (1, 5). What is the value of y on this curve when x = 2?
Approximating the area under a curve by summing the areas of rectangles is the basis of which idea?
Find ∫(3x² + 2) dx evaluated from x = 0 to x = 2.
What does the definite integral of a velocity function from t = a to t = b represent?
Find the area under y = 4x between x = 1 and x = 3.