With Shape Aware Quadratures (SAQ), for a given set of quadrature nodes, order and domain of integration, the quadrature weights are obtained by solving a system of suitable moment fitting equations in least square sense. The moments in the moment equations are approximated over a simplified domain that is a reasonable approximation of the original domain that are then corrected for the deviation of the shape of simplified domain from the original domain via shape correction factors. The shape correction factors can be derived based on a variety of sensitivity analysis techniques such as shape sensitivity and topological sensitivity. Using the right kind/order of shape correction factors for moment approximation enables the resulting quadrature (from the moment fitting equations) to efficiently adapt to the shape of the original domain even in the presence of numerous small features (such as holes, notches, and gaps). We demonstrate the efficacy of the method in integrating bivariate/trivariate polynomials over several 2D/3D domains in the presence of small features. SAQ is suitable for the computation of integrals arising in immersed boundary (IB) methods such as solution structure methods, fictitious domain methods, and Finite Cell Method (FCM). Hence, we study the efficacy of SAQ (in 2D/3D) to achieve a given accuracy in the context of FCM (one of the IB methods) which must perform numerical integration over arbitrary domains without meshing. SAQ automatically adapts to the shape of the original domain due to the incorporation of suitable shape correction factors leading to superior computational properties. In addition, SAQ offers a number of advantages including flexibility in the choice of quadrature points and basis functions that is usually not available for more traditional approaches. However, moment approximations in SAQ lead to error in the computed integral. Thus, there is need to bound/estimate this error. Hence, we present an a priori and a posteriori error estimate for SAQ (with SSA correction factor) based on Taylor's remainder theorem and higher-order Taylor series respectively. We also validate the error estimates using several computational examples in 2D.