Integral Calculus
Integral calculus questions test antiderivatives, the definite integral, and area. Bagrut problems ask you to compute an integral, find the area between two curves, or use the fundamental theorem of calculus to connect accumulation with rate of change.
Core rules
- ∫x^n dx=x^(n+1)/(n+1)+C for n≠−1; ∫(1/x)dx=ln|x|+C.
- ∫e^x dx=e^x+C, ∫cosx dx=sinx+C, ∫sinx dx=−cosx+C.
- Fundamental theorem: ∫_a^b f(x)dx=F(b)−F(a), where F′=f.
- Area between curves: ∫_a^b (upper−lower)dx, with intersection points as limits.
- A definite integral gives signed area; geometric area requires splitting where the curve crosses the axis.
- Substitution reverses the chain rule; change limits when using a definite integral.
Worked example
Find the area between y=x and y=x² from x=0 to x=1. The curves meet where x=x², so x=0 or x=1. On (0,1), x>x², so the integrand is x−x². Then ∫_0^1 (x−x²)dx=[x²/2−x³/3]_0^1=1/2−1/3=1/6. The subtraction order matters: reversing it gives −1/6, which is the negative of the area.
Traps
Omitting +C in an indefinite integral loses a mark. Using limits in the wrong order changes the sign. If the curves cross inside the interval, one integral does not give total area; split at the intersection. Do not integrate a product by integrating each factor separately.