ap calculus bc mcq practice test b\n81. let f and g be continuous functions for a ≤ x ≤ b. if a < c < b…

ap calculus bc mcq practice test b\n81. let f and g be continuous functions for a ≤ x ≤ b. if a < c < b, ∫ₐᵇf(x) dx = p, ∫_cᵇf(x) dx = q, ∫ₐᵇg(x) dx = r, and ∫_cᵇg(x) dx = s, then ∫ₐᶜ(f(x) - g(x)) dx =\n(a) p - q + r - s\n(b) p - q - r + s\n(c) p - q - r - s\n(d) p + q - r - s\n(e) p + q - r + s\no a\no b\no c\no d

ap calculus bc mcq practice test b\n81. let f and g be continuous functions for a ≤ x ≤ b. if a < c < b, ∫ₐᵇf(x) dx = p, ∫_cᵇf(x) dx = q, ∫ₐᵇg(x) dx = r, and ∫_cᵇg(x) dx = s, then ∫ₐᶜ(f(x) - g(x)) dx =\n(a) p - q + r - s\n(b) p - q - r + s\n(c) p - q - r - s\n(d) p + q - r - s\n(e) p + q - r + s\no a\no b\no c\no d

Answer

Explanation:

Step1: Use integral - property for splitting intervals

We know that $\int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx$. So, $\int_{a}^{c}f(x)dx=\int_{a}^{b}f(x)dx - \int_{c}^{b}f(x)dx=P - Q$.

Step2: Use integral - property for splitting intervals for $g(x)$

Similarly, $\int_{a}^{c}g(x)dx=\int_{a}^{b}g(x)dx-\int_{c}^{b}g(x)dx=R - S$.

Step3: Use integral - property for difference of functions

$\int_{a}^{c}(f(x)-g(x))dx=\int_{a}^{c}f(x)dx-\int_{a}^{c}g(x)dx$. Substitute the values from Step 1 and Step 2: $(P - Q)-(R - S)=P - Q - R+S$.

Answer:

B. $P - Q - R+S$